# A Simple, Fun, and Interesting Card Game

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I was skimming best-of list for games of 2018 and came across The Mind on several lists. It looks like a relatively simple, but interesting game, and after watching a couple of videos showing its gameplay, I decided that I could create more-or-less the equivalent thing just using a standard 52-card deck.

At first The Mind seems too simplistic; Players just cooperate to put out their cards in numerical order (I think it’s 1-100 in The Mind). What makes this hard is that you can’t talk to one another! So somehow you and the other player(s) have to “communicate” who should go next by … who knows … ESP or something? Under these circumstances, The Mind seems nearly impossible; how could you possibly know who’s got the next highest card without signaling in some way or other. I haven’t play The (original) Mind, but given that it gets such glowing reviews, I’m going to assume that what happens is that the players sort of learn one another’s non-verbal cues.

Regardless of how The (original) Mind works, I created what seems to be a very doable and fun version that uses a standard 52-card deck (plus jokers). Here’s how it goes:

Setup: Separate the (2) jokers, and the (13) hearts from the deck. Place the 2 jokers face up. These are your lives. Order the hearts from Ace-through-king face up, and place them in a stack, so you’ll be looking at an Ace. This is your level counter. (This is an “Ace Low” game: Ace = 1, J=11, Q=12, and K=13.) Shuffle the remaining 39 cards (the remaining 3 suits).

Overall Goal: The play is cooperative; when I say “you”, I mean, “you all the players cooperating”.  The hearts indicate your level, so you start at level 1 (ace), and win if you get to level 13 without dying either 2 or 4 times. (I recommend 4 times for the beginners). The jokers indicate your lives; Each time you fail, you either discard a joker if you’re playing 2-lives, or turn over, and then on the next fail discard a joker, thus affording you 4 lives. (Of course you can do life and level counting any way you like! This setup just makes things convenient.)

Level Play: For each level (1-13 == ace->king) play goes as follows:

1. Shuffle the 39 cards.
2. Deal as many cards to each player as your level. So on the first play each player gets one card, second level 2, etc. and if you should make it to the 13th (King) level, each player would get 13 cards. (There can only be 3 players, obviously, or else you’ll have to either play with multiple decks, or somehow otherwise modify the rules so that you don’t have to deal more cards than there are in the stack.)
3. Play as many rounds as you can get to without failing. If you fail, you decrement a life (jokers). If you get through all the dealt cards, then you win that level.
4. In either case, flip over one of the hearts to go on to the next level.
5. GoTo 1 🙂

Rounds within a level are really simple, but here is also where things get interesting:

Each round is independent — don’t worry about whether the cards between rounds are higher or lower than one another, HOWEVER, leave all the cards face up (you’ll see why this is in a moment).

Even though the players are cooperating, they cannot look at one another’s cards, nor talk to one another, except to decide which player should go first, then next, etc.

On each round, each player must place exactly one of their cards in an order that the players can discuss, but without telling one another what card they are playing. If the cards are placed in lowest-to-highest order, then that’s a win. If they are not, then the level is failed. (Ties count as wins; only out-of-order sequencing fails.)

For example, in level one, since each player has just one card, there will only be one round. If one of the players was dealt an Ace, they should argue strenuously to go first since regardless of what any other player could place, it will either tie or be larger than an Ace.

If you succeed at level 1, go on to level 2, where each player has 2 cards, and so on up to 13. If you haven’t failed 2 (or 4) times by the end of level 13, the game is won!

If you think this sounds either too easy or too hard to be interesting, try it and see for yourself; It’s actually quite difficult and interesting. Consider these factors:

1. As you get to higher levels, since there are only 3 of any card in the deck, you are more and more likely to be able to guess what cards the other players must have, especially combined with what has been already played. (Recall that I said it’s useful to leave all the played cards face-up and exposed. This helps in the card counting that can be a useful strategy.)
2. Since the rounds within a level are independent, you don’t have to play your lowest card on each play. It may be useful, for example, to play your very high cards early in order to get rid of some of them. Recall that you can’t talk about anything except who is going to place the next card, so you can’t agree no an algorithm. However, you can figure out, perhaps, what your team-mates’ algorithm might be, and play cooperatively with them based on this analysis.

So, there you go. Fun game for a rainy afternoon!

# Target Trig Tiff

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A few nights ago, Leo and I were playing with a glowing nerf bow-and-arrow-like rocket launching toy. I’m not much for toy weapons, but the “arrows” on this one are more like little rockets, they make a whizzing noise, and glow bright red, so it’s pretty fun at night in our cul-de-sac street.

Anyway, at one point Leo decided that he needed a bright white target, because it was getting quite dark, so he went inside and brought out a large sheet of art paper that he wanted me to hold up so he could shoot at it. Not really wanting to be the backstop for either his good or (more likely) poor aim, I suggested instead that it would be just as easy to lay the target on the ground.

This led to a bit of a tiff between us (the petty argument kind, not the tagged image file format kind!) Leo thought that laying it down on the street made it nearly impossible to target, whereas I thought it was more-or-less irrelevant. Later he depicted his argument with the following graphic:

That’s a bit hard to follow without the attached angry 9-year-old angrily explaining it at you, but the idea is that when the target is mounted on the wall (i.e., I’m holding it up, or it’s stuck to a tree, or something), as depicted in the center top and middle images, it is apparently larger than when it is laying on the ground. He made a rough guess about the ratio as the laying down apparent area being the old area divided by 2pi, or about 6x smaller.

On the left are depictions of how it’s harder to hit, in addition, because you have to shoot parabolically rather than directly at the target. I think that the the parabolic problem isn’t true — you’re always, shooting parabolically — although it may feel like you need to do that more-so because the thing is laying flat. So, I won’t pursue that part of the tiff. However, his point about the target’s apparent size is absolutely correct. Moreover, as he correctly, and angrily, pointed out, it’s worse for someone shorter!

I grasped the opportunity to redirect his ire into some trig, in order to figure out exactly how much small the target apparently is when laying down, as opposed to being mounted at eye level.

This is depicted here, and explained a little below. (I’ve slightly annotated the original page that Leo and I did the math on.)

We assumed a 6ft person (my eye level) looking at a 4ft target mounted at eye-level 20ft away. (For Leo, the observer would be about 1.5ft shorter.) We assumed as well that the target is laying flat with the center exactly at the 20ft point. We drew the two right triangles, for eye-to-far-end and eye-to-near-end of the fallen target (which would be at 22 and 18ft away along the base-line, respectively). Simple trig gave us the angles from vertical to eye-line for these, being 74.75 and 71.57 degrees, respectively.  Now it’s a little geometrically tricky, but arithmetically easy to compute the length of the vertically-projected line at 20ft between the two hypotenuses (hypothenusa? hypotenua? hypotena?), but breaking it into two right triangles that are similar to the larger ones. (This can probably be done by similar triangle ratios, but it was clearer, at least at that moment, to use trig.

The result is 1.211 ft. of apparent size, which isn’t 6x smaller, but it’s about 3x smaller. And although we didn’t compute it for Leo’s height, it would obviously have been even smaller than 1.2ft apparent size. (You can prove this to yourself by observing that if all you do is to turn the paper on edge, at 6ft, where it is mounted at eye-level, it will essentially disappear altogether because you’re looking at it edge on. In general, the more parallax you have, the larger it will appear. So a shorter person with less parallax will see a small apparent target size. So it may well be that it is ~6x smaller for someone 4″9′ tall!)

Now, actually what you want is not quite this, because one’s view goes around an arc, always at 90 degrees from the center. So there’s actually a small correction required to “lean” the page slightly, as depicted by the hypotenuse on to bottom right inset. I don’t show this computation, which I did later myself, but the final size comes to about 1.14ft of apparent size (as I recall), and this is actually still slightly off since I cheated by making a right triangle with one of the sight lines, but since we’ve been wildly rounding the angles all along, a greater level of precision is definitely not called for!

# Visualizing DNA? (Probably not!)

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Last week we did a week-long unit on ecosystem dynamics and evolution, which included the usual activities: looking at pond water microbes under the microscope, DNA extraction, walks in the woods (saw a snake!), predator-prey modeling, and so forth.

(Incidentally, we did our modeling in NetLogo, which is a terrific tool for teaching multi-agent modeling; I should do a separate post about this!)

Most this was run-of-the-mill science, and not too worthy of report, except for one interesting event.

DNA extraction from strawberries — using dish soap, salt, and alcohol — is about the most fun lab activity you can do in your kitchen! Here’s one of many all-the-same protocols you can find online. Unfortunately, all you get at the end is a huge clump of what is supposedly DNA. Since we were doing microscope work earlier in the week, I decided to try to visualize the DNA. My microscope goes up to about 470x (without oil), and since you need an electron microscope to see the helical structure at all, I figured that we wouldn’t see much in my light microscope.

However, when I was working in the lab we used a simple technique, called molecular combing, to stretch out the chromosomes in order to count them and get their sizes. I didn’t think that this would do anything useful in our kitchen, as you generally have to attach fluorescent markers to see the chromosomes in the lab, but I decided to give it a try anyhow.

Amazingly, just using the cover slip of the microscope slide to “comb” the strawberry DNA extract, we were able to visualize … well, I have no idea what this is:

They sure do look like helical structures of some sort. I’m guessing that these aren’t DNA, unless strawberry DNA are humongous! Rather, they are probably some sort of connective tissue from the strawberries. But for our first try at visualizing something DNA-like, it was pretty exciting. (I didn’t let on to Leo that these are almost certainly NOT DNA!)

Incidentally, this “photomicrograph” was taken by just putting my iPhone up to the eyepiece of the microscope. It took a few tries, but I’m slightly amazed that it worked at all!

# Raven Redux

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(I actually had to look up “redux” to make sure that it wasn’t a synonym of “reduce”.)

Anyway…

So, you probably don’t recall — because I hardly did — that about year ago Leo and I wrote a Lisp program to randomly generate “poems” by randomly selecting words from The Raven. Leo has recently been reading my Lisp book (heaven help him!), and got interested in revisiting our old program. Rather than just rehash it, I thought that we could do something a bit more clever this time around, I explained n-gram language models (although not in those terms), and then hash tables, with the goal of writing much more interesting poems.

Leo started out in Snuffy Code, which is a programming language he invented for himself some time ago, and which is actually pretty close to what computer scientists call pseudo-code. Here’s his Snuffy Code version of the algorithm:

You can see bits of our working on the data structures at the top, and to the right is a bit of our work on translating his Snuffy Code into Lisp. Here’s the Lisp:

(defvar *table* (make-hash-table :test #'equal))

(defun learn ()
(clrhash *table*)
(loop for sentence in *sentences*
do
(loop for (cur nxt) on (:start ,@sentence :end)
do (push nxt (gethash cur *table*)))))

(defun compose-line ()
(loop with cur = :start
as options = (gethash cur *table*)
as nxt = (nth (random (length options)) options)
until (eq nxt :end)
do (setq cur nxt)
collect cur))

(defun compose-poem ()
(loop for sentence in *sentences*
as this-length = (length sentence)
if (= this-length 0)
do (print nil)
else do (print (loop for random-line = (compose-line)
until (= this-length (length random-line))
finally (return random-line)))))

(In the previous post what is here called *sentences* was there called patterns, or p.)

All you need to do, then is:

(learn)
(compose-poem)

Et voila! (I’ve replaced the nils at paragraph breaks with double newlines.)

(DARKNESS PEERING LONG I PONDERED WEAK AND OMINOUS BIRD OR DEVIL)
(THEN THE RARE AND SAT ENGAGED IN THERE WONDERING FEARING)
(THIS EBONY BIRD AND THIS GRIM UNGAINLY FOWL TO BORROW)
(BY HORROR HAUNTED TELL THIS KIND NEPENTHE FROM AN UNSEEN CENSER)
(QUAFF OH QUAFF OH QUAFF OH QUAFF THIS MYSTERY EXPLORE)
(BUT THE NIGHTS PLUTONIAN SHORE)

(ON THE FACT IS THERE BALM IN THE NIGHTLY SHORE)
(WRETCH I PONDERED WEAK AND FOLLOWED FASTER TILL I SHRIEKED UPSTARTING)
(THOUGH THY MEMORIES OF A TAPPING AT MY FANCY INTO SMILING)
(FANCY UNTO FANCY THINKING WHAT THY FORM FROM AN UNSEEN CENSER)
(QUAFF OH QUAFF THIS I OPENED WIDE THE BUST AND STORE)
(TILL HIS CHAMBER DOOR)

(SHE SHALL CLASP A FEATHER THEN NO LIVING HUMAN BEING)
(IS SOMETHING AT MY HEART BE SHORN AND NOTHING MORE)
(NOTHING FARTHER THEN THIS AND BUST OF A FLIRT AND AN UNSEEN CENSER)
(OVER MANY A DEMONS THAT ONE WORD HE FLUTTERED)
(RESPITE AND CURIOUS VOLUME OF EVIL PROPHET SAID NEVERMORE)
(FOR THE LAMP-LIGHT GLOATED O ER)

(SIR SAID I NODDED NEARLY NAPPING SUDDENLY THERE SPOKEN)
(TIS THE PLACID BUST ABOVE HIS SONGS ONE GENTLY RAPPING)
(STARTLED AT MY BOOKS SURCEASE OF LORD OR BEAST UPON THE RAVEN NEVERMORE)
(ON THE RAVEN OF THE SAINTLY DAYS OF LORD OR DEVIL)
(PERCHED UPON A SAINTED MAIDEN WHOM THE SCULPTURED BUST OF THE NIGHTS PLUTONIAN SHORE)
(BACK THE NIGHTS PLUTONIAN SHORE)

(AS A MINUTE STOPPED OR BEAST UPON THE RAVEN OF YORE)
(THIS EBONY BIRD OF SORROW FOR THE COUNTENANCE IT WORE)
(TELL ME TELL ME AS OF BIRD BEGUILING MY CHAMBER DOOR)
(NOT A TAPPING TAPPING TAPPING AT MY HOPES HAVE FLOWN BEFORE)
(BUT THE SILENCE WAS UNBROKEN QUIT THE BIRD OF FORGOTTEN LORE)
(THEN METHOUGHT THE TUFTED FLOOR)

(BY THE SHUTTER WHEN WITH THE GRAVE AND THE CHAMBER DOOR)
(SWUNG BY THAT LIES FLOATING ON THE LAMP-LIGHT O ER)
(TIS THE SHUTTER WHEN WITH SORROW LADEN IF HIS CHAMBER DOOR)
(SHE SHALL CLASP A DEMONS THAT IS DREAMING DREAMS NO LONGER)
(SOME ONE GENTLY RAPPING RAPPING AT MY HEART BE LIFTED NEVERMORE)
(THAT IS ITS ONLY WORD LENORE)

(DESOLATE YET ALL THE ONLY THIS AND THIS KIND NEPENTHE AND THE FLOOR)
(FOR THE SILENCE WAS SURE I HEARD YOU CAME A TAPPING TAPPING)
(FANCY THINKING WHAT THIS SOUL WITHIN THE TEMPEST TOSSED THEE HERE I IMPLORE)
(SIR SAID I WHEELED A RARE AND THE STILLNESS GAVE NO CRAVEN)
(TIS SOME VISITOR I PONDERED WEAK AND THIS DESERT LAND ENCHANTED)
(NOT A TOKEN OF FORGOTTEN LORE)

(SHALL PRESS AH DISTINCTLY I HAD SOUGHT TO DREAM BEFORE)
(IN THE CUSHIONS VELVET SINKING I WISHED THE LAMP-LIGHT O ER)
(BUT THE FOWL WHOSE FIERY EYES NOW BURNED INTO THE LAMP-LIGHT GLOATED O ER)
(AH DISTINCTLY I CRIED THY SOUL WITH SUCH NAME LENORE)
(QUOTH THE COUNTENANCE IT IS SITTING STILL IF WITHIN THE NIGHTLY SHORE)
(AND SO APTLY SPOKEN)

(IT UTTERS IS I HEARD A BUST SPOKE ONLY WORD LENORE)
(WRETCH I SAT ENGAGED IN THE MORROW HE)
(TIS SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(SOME VISITOR ENTREATING ENTRANCE AT MY SOUL IN FRONT OF LENORE)
(CAUGHT FROM SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(OF A STATELY RAVEN NEVERMORE)

(QUOTH THE SHUTTER WHEN WITH MY HEART BE THAT IS DREAMING)
(BUT THE WHISPERED AND NOTHING FARTHER THEN HE UTTERED NOT A BUST OF YORE)
(SWUNG BY REPLY SO FAINTLY YOU CAME RAPPING AT EASE RECLINING)
(THAT NOW BURNED INTO THAT IS ON THIS AND NOTHING MORE)
(DESOLATE YET ALL THE AIR GREW DENSER PERFUMED FROM THE ANGELS NAME LENORE)
(THAT NOW TO DREAM BEFORE)

(TIS SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(AND THE RAVEN STILL A FEATHER THEN THE LAMP-LIGHT GLOATING O ER)
(GET THEE BY THE SEEMING OF THE FLOOR)
(AND BUST AND RADIANT MAIDEN WHOM THE BIRD BEGUILING MY DOOR)
(TIS SOME VISITOR ENTREATING ENTRANCE AT MY HEART AND MORE)
(PRESENTLY MY DOOR)

(TO THE ONLY STOCK AND THIS AND AN UNSEEN CENSER)
(SHALL PRESS AH DISTINCTLY I WHISPERED AND FOLLOWED FASTER TILL THE WHISPERED WORD LENORE)
(FANCY UNTO FANCY THINKING WHAT THEREAT IS AND NOTHING MORE)
(DOUBTLESS SAID I WHEELED A BUST ABOVE HIS CHAMBER DOOR)
(FOLLOWED FASTER TILL I THING OF SOME VISITOR I SHRIEKED UPSTARTING)
(LEAVE MY CHAMBER DOOR)

(THEN UPON A SAINTED MAIDEN WHOM THE ANGELS NAME LENORE)
(BUT WHOSE FOOT-FALLS TINKLED ON THIS EBONY BIRD OF EACH PURPLE CURTAIN)
(AND NEPENTHE FROM THY SOUL FROM MY SOUL WITH THE RAVEN SITTING)
(SIR SAID I SCARCE WAS UNBROKEN QUIT THE NIGHTS PLUTONIAN SHORE)
(TIS SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(BE SHORN AND NOTHING MORE)

(THRILLED ME WITH MY HEART BE STILL IF BIRD OF LENORE)
(MEANT IN THAT SHADOW ON THE SAINTLY DAYS OF YORE)
(MEANT IN GILEAD TELL ME AS IF WITHIN THE RAVEN STILL IS AND SO PLAINLY)
(BIRD OR FIEND I FLUNG THE NIGHTS PLUTONIAN SHORE)
(WHILE I STOOD THERE CAME A MINUTE STOPPED OR STAYED HE)
(THIS DESERT LAND ENCHANTED)

(WRETCH I WISHED THE FACT IS ON THE RAVEN NEVER FELT BEFORE)
(EVER YET WAS IN GUESSING BUT THE NIGHTS PLUTONIAN SHORE)
(PERCHED UPON THE TEMPEST AND STERN DECORUM OF LENORE)
(TELL ME I HEARD YOU CAME TAPPING AT MY CHAMBER DOOR)
(MERELY THIS SOUL WITH MY CHAMBER TURNING ALL THE ONLY WORD THERE SPOKEN)
(THIS DESERT LAND ENCHANTED)

(BY HORROR HAUNTED TELL ME TRULY YOUR FORGIVENESS I REMEMBER IT WORE)
(WHAT THY BEAK FROM AN ECHO MURMURED BACK INTO THE NIGHTS PLUTONIAN SHORE)
(BUT THE FACT IS SITTING LONELY ON THE NIGHTS PLUTONIAN SHORE)
(MEANT IN GILEAD TELL THIS GRIM UNGAINLY FOWL TO DREAM BEFORE)
(THIS GRIM UNGAINLY FOWL WHOSE VELVET-VIOLET LINING THAT THE MORROW HE)
(RESPITE AND SO PLAINLY)

(THEN METHOUGHT THE GRAVE AND FORGET THIS I THING OF MY WINDOW LATTICE)
(AS A RARE AND CURIOUS VOLUME OF PARTING BIRD OR DEVIL)
(TIS THE ONLY WORD AS A MIDNIGHT DREARY WHILE I WAS THE RAVEN NEVERMORE)
(RESPITE AND FORGET THIS HOME BY THESE ANGELS HE FLUTTERED)
(AND FOLLOWED FAST AND SO THAT ONE WORD THERE CAME RAPPING AT MY CHAMBER DOOR)
(THEN THE RAVEN NEVERMORE)

(LEAVE ME I SAID I NODDED NEARLY NAPPING SUDDENLY THERE SPOKEN)
(TELL THIS I STOOD THERE CAME A QUAINT AND NOTHING MORE)
(FOLLOWED FAST AND THE PALLID BUST OF HIS SOUL FROM THE DISTANT AIDENN)
(EAGERLY I MUTTERED TAPPING SOMEWHAT LOUDER THAN MUTTERED OTHER FRIENDS HAVE FLOWN BEFORE)
(BUT WITH MIEN OF PALLAS JUST ABOVE US BY THAT GOD HATH SPOKEN)
(ONCE UPON THE FLOOR)



Before going any father I must point out some of the gems from this run, and a few others from other runs of the same code:

(QUAFF OH QUAFF OH QUAFF OH QUAFF THIS MYSTERY EXPLORE)
(FANCY UNTO FANCY THINKING WHAT THY FORM FROM AN UNSEEN CENSER)
(TIS SOME VISITOR ENTREATING ENTRANCE AT MY HEART AND MORE)
(PROPHET STILL IS SOMETHING AT MY LONELINESS UNBROKEN QUIT THE RAVEN OF LENORE)
(RESPITE RESPITE RESPITE RESPITE RESPITE RESPITE RESPITE RESPITE RESPITE RESPITE AND FLUTTER)
(QUOTH THE CUSHIONS VELVET LINING WITH FANTASTIC TERRORS NEVER NEVERMORE)

and my personal favorite:

(THOUGH ITS ANSWER LITTLE MEANING LITTLE MEANING LITTLE MEANING LITTLE MEANING LITTLE RELEVANCY BORE)

That was fun. After Leo went off to other things I decided to do a little refinement of my own, so I did a trivial little hack to improve the flow a little. First I rewrote the data by separating the paragraphs (whereas, in *sentences* there’s no concept of a paragraph, except for the nils between them.) That is, I created this structure:

(defparameter *paragraphs*
'(
(
(Once upon a midnight dreary  while I pondered  weak and weary  )
(Over many a quaint and curious volume of forgotten lore  )
(    While I nodded  nearly napping  suddenly there came a tapping  )
(As of some one gently rapping  rapping at my chamber door  )
( Tis some visitor  I muttered  tapping at my chamber door  )
(            Only this and nothing more  )
)
(
(    Ah  distinctly I remember it was in the bleak December  )
(And each separate dying ember wrought its ghost upon the floor  )
(    Eagerly I wished the morrow  vainly I had sought to borrow )
(    From my books surcease of sorrow sorrow for the lost Lenore  )
(For the rare and radiant maiden whom the angels name Lenore  )
(            Nameless here for evermore  )
)
...and so on...
))

Then, by integrating learning and composition, and then simply localizing the global *sentence*, which both the learning and composition use, first to 3-paragraph windows for learning, and the to each paragraph for composition:

(defun compose-progressive-poem ()
(loop for (p1 p2 p3) on *paragraphs*
do (let ((*sentences* (append p1 (append p2 p3)))) (learn))
(print '==============)
(let ((*sentences* p1))
(compose-poem))))

we get much better conceptual localization:

(EAGERLY I HAD SOUGHT TO STILL THE RARE AND NOTHING MORE)
(AH DISTINCTLY I WISHED THE BEATING OF EACH PURPLE CURTAIN)
(THIS IT IS AND CURIOUS VOLUME OF EACH PURPLE CURTAIN)
(WHILE I REMEMBER IT WAS IN THE BEATING OF FORGOTTEN LORE)
(TIS SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(AH DISTINCTLY I STOOD REPEATING)
==============
(TIS SOME LATE VISITOR ENTREATING ENTRANCE AT MY CHAMBER DOOR)
(THIS IT WAS SURE I SCARCE WAS NAPPING AND NOTHING MORE)
(NAMELESS HERE I SCARCE WAS SURE I HAD SOUGHT TO BORROW)
(NAMELESS HERE I IMPLORE)
==============
(PRESENTLY MY HEART I SCARCE WAS UNBROKEN AND NOTHING MORE)
(AND SO FAINTLY YOU CAME TAPPING AT MY CHAMBER DOOR)
(DEEP INTO THAT I WHISPERED AND SO FAINTLY YOU HERE I STOOD REPEATING)
(MERELY THIS IT IS I HEARD YOU CAME RAPPING)
(THIS IT IS I STOOD THERE AND NOTHING MORE)
(AND SO FAINTLY YOU CAME RAPPING)
==============
(PRESENTLY MY SOUL WITHIN ME SEE THEN NO TOKEN)
(TIS THE WIND AND THE SILENCE WAS THE CHAMBER DOOR)
(BACK THE SILENCE WAS SURE I OPENED WIDE THE WHISPERED AND NOTHING MORE)
(BUT THE STILLNESS GAVE NO MORTAL EVER DARED TO DREAM BEFORE)
(DEEP INTO THE FACT IS I SCARCE WAS SURE I SURELY THAT I IMPLORE)
(DOUBTING DREAMING DREAMS NO TOKEN)
==============
(NOT THE STILLNESS GAVE NO MORTAL EVER DARED TO DREAM BEFORE)
(LET MY CHAMBER TURNING ALL MY SOUL WITHIN ME BURNING)
(DEEP INTO THAT DARKNESS PEERING LONG I WHISPERED AND NOTHING MORE)
(TIS THE LEAST OBEISANCE MADE HE NOT THE ONLY WORD LENORE)
(DEEP INTO THE SAINTLY DAYS OF THE WHISPERED AND NOTHING MORE)
(TIS THE WHISPERED WORD LENORE)
==============
(THEN WHAT THY LORDLY NAME IS SOMETHING AT MY CHAMBER DOOR)
(PERCHED AND ANCIENT RAVEN WANDERING FROM THE NIGHTS PLUTONIAN SHORE)
(THOUGH THY CREST BE STILL A TAPPING SOMEWHAT LOUDER THAN BEFORE)
(TELL ME SEE THEN THIS EBONY BIRD BEGUILING MY WINDOW LATTICE)
(SOON AGAIN I SAID I SURELY THAT IS AND NOTHING MORE)
(BY THE LEAST OBEISANCE MADE HE)
==============
(PERCHED UPON A MINUTE STOPPED OR BEAST UPON THE SAINTLY DAYS OF YORE)
(NOT A FLIRT AND ANCIENT RAVEN OF THE SCULPTURED BUST OF YORE)
(IN THERE STEPPED A MINUTE STOPPED OR LADY PERCHED ABOVE MY CHAMBER DOOR)
(TELL ME WHAT THY LORDLY NAME IS ON THE COUNTENANCE IT WORE)
(WITH MIEN OF LORD OR LADY PERCHED ABOVE HIS CHAMBER DOOR)
(PERCHED UPON THE NIGHTS PLUTONIAN SHORE)
==============
(THOUGH THY LORDLY NAME AS MY SAD FANCY INTO SMILING)
(THEN THIS EBONY BIRD SAID ART SURE NO LIVING HUMAN BEING)
(THEN THE RAVEN SITTING LONELY ON THE MORROW HE WILL LEAVE ME AS NEVERMORE)
(FOR WE CANNOT HELP AGREEING THAT ONE WORD AS NEVERMORE)
(THOUGH THY CREST BE SHORN AND STERN DECORUM OF THE RAVEN NEVERMORE)
(BY THE NIGHTLY SHORE)
==============
(FOLLOWED FASTER TILL HIS SOUL IN THAT NO LIVING HUMAN BEING)
(CAUGHT FROM SOME UNHAPPY MASTER WHOM UNMERCIFUL DISASTER)
(FOLLOWED FAST AND FOLLOWED FAST AND FOLLOWED FAST AND STORE)
(TILL I MARVELLED THIS UNGAINLY FOWL TO HEAR DISCOURSE SO PLAINLY)
(TILL I SCARCELY MORE THAN MUTTERED OTHER FRIENDS HAVE FLOWN BEFORE)
(NOTHING FARTHER THEN HE FLUTTERED)
==============
(ON THE MORROW HE UTTERED NOT A FEATHER THEN HE FLUTTERED)
(ON THE MORROW HE UTTERED NOT A CUSHIONED SEAT IN THAT MELANCHOLY BURDEN BORE)
(ON THE RAVEN STILL BEGUILING ALL MY HOPES HAVE FLOWN BEFORE)
(TILL I SCARCELY MORE THAN MUTTERED OTHER FRIENDS HAVE FLOWN BEFORE)
(FOLLOWED FASTER TILL I WHAT IT UTTERS IS ITS ONLY STOCK AND DOOR)
(FANCY UNTO FANCY INTO SMILING)
==============
(STARTLED AT THE STILLNESS BROKEN BY REPLY SO APTLY SPOKEN)
(STRAIGHT I WHEELED A CUSHIONED SEAT IN FRONT OF BIRD OF YORE)
(WHAT THIS AND OMINOUS BIRD OF NEVER NEVERMORE)
(WHAT THIS I SAT DIVINING WITH MY HEAD AT EASE RECLINING)
(ON THE FOWL WHOSE FIERY EYES NOW BURNED INTO SMILING)
(OF NEVER NEVERMORE)
==============
(BUT WHOSE FOOT-FALLS TINKLED ON THE LAMP-LIGHT GLOATED O ER)
(THIS I BETOOK MYSELF TO THE FOWL WHOSE FOOT-FALLS TINKLED ON THE TUFTED FLOOR)
(THIS GRIM UNGAINLY GHASTLY GAUNT AND OMINOUS BIRD OF LENORE)
(WRETCH I BETOOK MYSELF TO THE LAMP-LIGHT GLOATING O ER)
(ON THE CUSHIONS VELVET SINKING I CRIED THY MEMORIES OF YORE)
(BUT NO SYLLABLE EXPRESSING)
==============
(WRETCH I SAT DIVINING WITH THE LAMP-LIGHT GLOATING O ER)
(QUOTH THE FOWL WHOSE FIERY EYES NOW BURNED INTO MY BOSOMS CORE)
(RESPITE AND MORE I SAT ENGAGED IN GUESSING BUT NO SYLLABLE EXPRESSING)
(RESPITE RESPITE RESPITE RESPITE RESPITE AND MORE I THING OF LENORE)
(WRETCH I SAT ENGAGED IN GUESSING BUT NO SYLLABLE EXPRESSING)
(SHE SHALL PRESS AH NEVERMORE)
==============
(WRETCH I CRIED THY MEMORIES OF EVIL PROPHET SAID I IMPLORE)
(CLASP A RARE AND RADIANT MAIDEN WHOM THE RAVEN NEVERMORE)
(PROPHET SAID I CRIED THY MEMORIES OF EVIL PROPHET STILL IF WITHIN THE RAVEN NEVERMORE)
(RESPITE RESPITE AND RADIANT MAIDEN WHOM THE RAVEN NEVERMORE)
(QUAFF OH QUAFF OH QUAFF OH QUAFF THIS DESERT LAND ENCHANTED)
(QUOTH THE RAVEN NEVERMORE)
==============
(IS THERE IS THERE BALM IN GILEAD TELL THIS DESERT LAND ENCHANTED)
(PROPHET SAID I THING OF EVIL PROPHET SAID I IMPLORE)
(CLASP A SAINTED MAIDEN WHOM THE NIGHTS PLUTONIAN SHORE)
(QUOTH THE TEMPEST TOSSED THEE BACK INTO THE ANGELS NAME LENORE)
(TELL ME TELL THIS SOUL WITH SORROW LADEN IF WITHIN THE RAVEN NEVERMORE)
(QUOTH THE DISTANT AIDENN)
==============
(AND MY HEART AND TAKE THY FORM FROM OFF MY CHAMBER DOOR)
(TAKE THY SOUL WITH SORROW LADEN IF BIRD OR FIEND I SHRIEKED UPSTARTING)
(TELL THIS SOUL FROM OFF MY SOUL FROM OFF MY DOOR)
(PROPHET SAID I THING OF EVIL PROPHET SAID I SHRIEKED UPSTARTING)
(SHALL CLASP A DEMONS THAT HEAVEN THAT GOD WE BOTH ADORE)
(AND MY CHAMBER DOOR)
==============
(LEAVE NO BLACK PLUME AS A DEMONS THAT SHADOW ON THE RAVEN NEVERMORE)
(AND HIS SHADOW ON THE RAVEN NEVER FLITTING STILL IS DREAMING)
(AND THE LAMP-LIGHT O ER HIM STREAMING THROWS HIS SHADOW ON THE RAVEN NEVERMORE)
(BE THAT LIE THY BEAK FROM OFF MY CHAMBER DOOR)
(AND THE LAMP-LIGHT O ER HIM STREAMING THROWS HIS SHADOW ON THE NIGHTS PLUTONIAN SHORE)
(AND THE RAVEN NEVERMORE)
==============
(ON THE PALLID BUST OF PALLAS JUST ABOVE MY CHAMBER DOOR)
(AND THE RAVEN NEVER FLITTING STILL IS SITTING STILL IS SITTING)
(AND HIS SHADOW ON THE SEEMING OF PALLAS JUST ABOVE MY CHAMBER DOOR)
(AND HIS EYES HAVE ALL THE SEEMING OF A DEMONS THAT IS DREAMING)
(AND THE PALLID BUST OF A DEMONS THAT LIES FLOATING ON THE FLOOR)
(SHALL BE LIFTED NEVERMORE)`

# NP Complete 3rd Grade Math Problems

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Leo’s third grade class got to try a Noyce Foundation math worksheet [1] the other day. They didn’t fare so well, but I gotta tell you, some of the problems are REALLY hard! In fact, one of them is NP-Hard! Now, I’m sure that there must be a trick to this problem, but I wasn’t able to see it in the couple of minutes that I thought about it. In fact, Leo even recognized that it’s NP-Hard; He told me the day of the quiz that they had given out an NP-Complete problem! (In fact, if you look closely at the page, you’ll see that he wrote “NP Compleite” (Sic 🙂 ) diagonally across the picture:

One of the things I’ve told my students over the years, only half jokingly, is that computer scientists are fundamentally lazy; Why should we work when we can get a computer to work for us?! Since I couldn’t figure out off the top of my head how to solve this problem without trying all possible combinations (and god help you if you allow repeats!), Leo and I set about writing a program to solve it for us.

Without going into great detail, here’s the code:

*Items* is the list of items as pairs, as: (name . price):

The only interesting function here is the combs function, and that’s the only one that I bothered to work through in detail with Leo. We first set out the recurrence on the board (bottom to top, excluding the top line, which is the input):

Leo got the recurrence pattern right away, although I, of course, had to help him put it into Lisp. (Compare with his Snuffycode version, below.)

Of course, first making a huge list of all possible combinations, and then scoring each one, is extremely space-inefficient. A depth first search would be better, but making all the combinations first is conceptually simpler.

Anyway, space-wasting aside, calling it (via seek\$) results in 279 non-redundant solutions.

Here are the first few:

We also found the shortest and longest solutions (shortest was 7 — the reduce…mapcar got cropped out of the screencap):

After all this was done, Leo wanted to write the program in “Snuffycode”, his own private programming language (for which, thank goodness, neither a compiler or interpreter exists):

Note that this code uses a different search method, counting up to 2^20, and using binary expansion to select the item list. Snuffycode, being a bit like APL, has implicit type conversion (L=N), operators that select a subset of an array when the array index is represented in binary (A[L]), and an operator that sums up arrays (/ \). 🙂

By the way, Leo says that some kids actually solved the problem exactly, so there may be a trick that wasn’t obvious to Leo or me, or maybe they just got lucky. It seems to me that there must be a trick, because if you’re only looking for 1 solution in 279 out of 2^20, that’s about a 1 in 3000+ chance of finding it by luck. (The problem might be easier is you allow redundant solutions. Although that makes for many more possible solutions if you were brute-forcing it (technically, an infinite number! … although you could apply a sensible limit), you maybe could do some sort of stacking up to a target that you can then fill in….or some such algorithm.

Like I said at the outset, it’s not worth thinking deeply about such a small problem; That’s why we invented computers!

By the way, the problems in the Noyce set were supposedly in difficulty order, and this was only the second of five problems! I solved the third one using straightforward algebra (although apparently some of Leo’s classmates solved it by brute force — remember, these are THIRD GRADERS!). The fourth required slightly more complex algebra.

The fifth problem was extremely easy, if you know anything about probability.

Go figure…

[1] Unfortunately, I think that the Noyce Foundation has ceased operations.

# The Dragon Box Algebra Teaching Game

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Over the past couple of weeks, Leo and I have worked through the entire Dragon Box Algebra 12+ app. Dragon Box builds a series of gamified math apps. I can’t speak to any of the others, but Algebra 12+ (hereafter, DBA12+) is about as fun as I could imagine making Algebra, unless, like me, you think that Algebra is just inherently fun on its own! Unfortunately, DBA12+ has, to my mind, some significant flaws.

Note that Leo is only 8 (closer to 9), and I’m not sure what the “12+” means — probably target age — so I hadn’t expected the thing to be self-usable by an 8/9 year old. So Leo and I worked every problem together, although I tried to do my best hands-off scaffolding by only intervening when he was absolutely stuck — although sometimes I was absolutely stuck as well! Another thing that I did was to provide running commentary, giving names, and sometimes explanations, to back up the algebraic concepts being demonstrated, or those he was employing in play. In  my parent-child interaction research we called this “active language”, and we hypothesized that this matters…a lot! I’ll get back to this below.

First some of the great things about the game. The central conceit and motivation is well designed and fun: You are feeding a creature in a box to make it grow from an egg into a big scary (in a fun-scary sense) creature. (The box is your algebraic unknown — generally “X”, as will be eventually revealed.) The artwork is great, and the goal is motivating (at least to an 8/9 year old). The moldularization and “leveling up” of the algebra content is well done too. There are around 10 major levels, each with 20 problems. New capabilities, like factoring, are introduced usually at the major level breaks, and new equation-level structural complexities are introduced within levels. This works extremely well, and the problems are extremely well designed to very rapidly bring together and reinforce previously learned capabilities into more and more complex problems, without being either boringly redundant, nor taking too large steps. Leo (sometimes with my help around the edges) was able to “solve” pretty much every problem, although the latter problems sometimes took a lot of exploration — sometimes even blind trial-and-error.

One thing that DBA12+ does that is, one the one hand very clever, but on the other hand somewhat annoying, is to begin with entirely non-mathematical symbols, specifically, pictures of funny creatures, and then, as you level up, it introduces more and more mathematical notation. Sometimes this works, and sometimes it doesn’t. I think that the creatures are supposed to represent symbols that you might see in an algebraic expression, like A, B, M, N, and Y, and the box is always definitely X. But soon enough numbers are introduced (as dice-sides, initially, as with dots, and eventually as numbers), but there are operations that work on numbers that don’t work on symbols, like factoring, so you sort of have to figure this out by trial and error, because the animals could just as easily have been representing numbers!

There are some unfortunate quirks imposed by a combination of the gamification and the requirements of the leveling-up educational sequencing. For example, until very late in the game you are only allowed to add/subtract/multiply/divide through by “reified” terms that it offers you at the bottom of the screen. This blocks some paths that would make sense (to me at least), and which would potentially provide simpler solutions. In the next-to-last level of the whole game you are given an operator that will make (almost) any term in the current equation into a “reified” term, and only at that point can you do what we had been reaching for throughout the game to that point!

Another pretty painful aspect of DBA12+ is that, although there is a “help” button, it only gives you the complete solution, not hints nor steps that you might consider. So if you’re in the middle of a solution, it’s not smart enough to give you a hint from where you are; You just have to follow it back to its one-and-only-one “correct” solution. (Their solutions are actually often more complex than necessary. In about a quarter of the cases we were able to find solutions that are shorter than theirs!)

One significant problem is that what DBA12+ counts as a “correct” solution is sometimes a bit hard to fathom. A target number of steps is given for solving the problem, and if you go over those, you don’t get “full credit” (three stars). This has the positive effect of tending against “random” solutions where you just keep moving things around until you get something, but it also tends against a very useful strategy, which is to simply move things around until you get a solution, then you’ll know what the solution is, and can start over and work on a more efficient path.

Also, and very unfortunately, like many other games, once you “try”, whether you are correct or not, the solution vanishes into the next problem, so you don’t really get a chance to think about your solution, or their solution, or pretty much anything at all. There’s actually no way to think about your solution, because there’s no way to review it. This would be a nice addition: A way to look at the steps you took to get to wherever you find yourself, whether right, wrong, or in the middle of the solution, and esp. at the end!

Another aspect of DBA12+’s obscurity with regard to “correct” solutions is there there are certain forms that it is expecting as “the correct answer”, and others that are formally correct (because, of course, you’re only allowed to correctly apply algebraic rules), but which are somehow, to DBA12+’s mind, not simplified enough … or something. Because there are no words in the thing, except for the work “Yuk”, when it doesn’t like your solution, it can’t explain to you why it doesn’t like certain solutions. So you are sometimes — quite often, in fact — left guessing as to what it didn’t like about your solution, and just trying different things randomly to try to get to a solution it likes. In a similar vein, when there’s just a singleton box left on either side of the equation, it considers you to have completed the problem, and tries to “eat” everything on the other side, like it or not! I would have preferred, for all of the above reasons, a button called “Try” (or some-such name or appropriate icon) which, when pressed, the dragon comes out of the box and tries your solution, and if it doesn’t like it, you get to continue working, or back out, etc. And it would be great if, when it didn’t like something you left it, it told you why.

This leads me to my main criticism of DBA12+, which is that there are no explanations at all at any time, no explanations of why it doesn’t like your solution, and no naming of operations. To my mind, this significantly diminishes the educational utility of the product. If I was designing this app, as new operations were introduced, I would provide “active language”, giving names to the operations, as I was over Leo’s shoulder, like factoring, dividing or multiplying through, and so forth. And I would explain the constraints on when you can and can’t do these in terms that made sense to him in the context, using the vocabulary we had built up. Unless you have someone who knows Algebra pretty well sitting over your shoulder, there’s just no way you can make sense of some of the constraints that are either inherent in algebra itself (like that you can’t add fractions that don’t have a common denominator), or those imposed by DBA12+, like you can only do integer arithmetic (including rationals via fractions), and can only factor certain things and not others, or those leading it to disliking certain solutions that it simply thinks are “Yuk”y.

If, when new capabilities are introduced it would give them names, like factoring or distribution, and then these were maybe flashed on the screen whenever you used them, it would enable so much more depth of understanding of what’s going on. For example, you might be able to tell that factoring a common term out of a parenthesized expression is closely related to factoring a number, or that the “through” operations (dividing through, etc.) are related in terms of their application across all the terms. It could also use these terms in hints (e.g., “Maybe consider factoring?”), and in explanations for Yukky (Yuky?) results (e.g., “This isn’t in its simplest form!”)

After we went through the whole game, I had the opportunity to test Leo’s understanding of algebra in a natural experiment that happened to arise when we were talking about the earth’s polar regions. He asked why “they chose” -40 to be the same in Fahrenheit and Centigrade” (His hypothesis was that the folks who invented temperature, or perhaps just centigrade, lived in a polar region! 🙂 ) Aha! A perfect opportunity for some algebra! (The benighted reader can easily work this out for him or herself by taking the equation: C=5/9(F-32), or whatever version of that you like, asserting the fixed point: C=5/9(C-32), and then solving for C.) I used the DBA12+ conceit by making the C into a box, and asking him to solve the equation, just like in DBA12+. There are, of course, many paths to get to C=-40, but the DBA12+ way, where you can only do integer arithmetic, and can’t distribute composite terms (like 5/9) is to start from C=(5(C-32))/9, multiply through by 9, yielding 9C=5(C-32), distribute the 5 to get 9C=5C-(5*32), isolate: 9C-5C=-(5*32), do the math (on both sides): 4C=-(160), simplify out the minus sign (could be done earlier, or by distribution, or just drop the parens): 4C=-160, and then divide through by 4 to get just C=-40.

Notice how nice it is to express this solution path in a combination of words and symbols, and how if we were going to have a conversation about it, we’d be diminished to grunting and pointing if we didn’t have names for things! Active Language provides these names and ways of speaking about activity, and through this process forms the foundation for reasoning. Human language is possibly our only unique feature, and it is probably the most ennobling and enabling! Unfortunately, by trying to force absolutely everything into fun gestures and graphics, DBA12+, and similar “educational” games, diminishes their topic —  algebra in this case — to mere grunting and pointing.

# Portal Physics

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Elsewhere I’ve briefly written about Leo’s obsession with Portal, a clever, (mostly) non-violent, Steam game. As always, I try to turn his obsessions into lessons (I should trademark that!) and so it is with Portal. In another post I talked about the game’s clever “AIs” (scare-quoted because, of course, they are merely scripted, not real AIs!) Here I talk about a cool little physics lesson that we recently did using Portal.

One of the really fun and clever things that you can do with portals is to put an In-Portal in a floor, and an Out-Portal in a wall, and then drop something into the In-Portal and it’ll come out of the wall. If you drop it from a distance into the In, it’ll blow out the Out with what we shall hypothesize is the same horizontal velocity at which is hit the In in the floor.(Watch this happen quite dramatically at about 4:40 in this video.)

This can be quickly turned into an interesting physics question: How high do I have to jump from such that I go a specified distance across the room? Here’s the setup (I’ve redrawn all this because we initially did it on a white board, and between erasures and scribbling, the white board is entirely incomprehensible!)

A box is dropped (or a person jumps!) from the tube at the top left at a height of Hf, and accelerates (presumably at 9.8m/s^2), hitting the In portal (the circle at the bottom left) with Vf. It instantaneously blows out of the Out portal (the circle on the wall in the middle left of the picture, at a height of Hp) with the same velocity, Vf and immediately begins to fall, eventually hitting the floor and stopping at distance D. (We neglect a lots of the usual things like rolling along the floor, air resistance, and that the box isn’t a point mass.)

The question is: How high to I need to put the Out portal (Hp), if the drop point (Hf) is, say, 20 meters high, and I want the box to land (D) at, say, 20meters from the left wall?

Notice that there are three equations involved here [all equations from this wikipedia page, or any of the other zillion pages that have the same info]: For the drop we can compute Vf from Hf (20m) (given g=9.8m/s^2). But in order to figure out how far it’s going to go to the right after coming out of the wall need to compute for how LONG (i.e., t in secs) we want it to travel in order to hit the ground in D (20m), and then back-compute the high (Hp) given how long we want it to be in the air before hitting the ground.

Okay so $V_f=V_0+{\sqrt{2gH_f}}$ [Since the drop is static, $V_0=0$.] We’ll round 9.8 to 10m/s^2, so $V_f={\sqrt{20*10*20}}={\sqrt{20^2}}$, which, coincidentally, is really easy to calculate: Vf=20m/s.

Okay, now at 20m/s going 20m upon exit from the Out portal, leads to the sort of trivial use of $t=D/V_f=20/20$, which is exactly 1 sec. (I actually didn’t make these numbers up to come out so nicely!)

Finally, we need to figure out how far something will fall in 1 second, which is just $1/2*g*t^2$, where t=1, and g=10, as above, so $H_p=1/2*10*1 = 5$ meters. So, we have to put the (center of the) portal 5 meters up the left wall.

By a little algebra we can reduce these three equations:

$V_f={\sqrt{2gH_f}}$
$t=D/V_f$
$H_p=1/2*g*t^2$

into one:

$H_p={gD}/({2\sqrt{2gH_f}})$

I’ll update this post with a screen cap from Leo’s actually implementing this so check it out (spoiler alter: it worked out pretty well!)

[By the way, there are many quite clever videos where people work out aspects of the physics of Portal, as well as many funny thought experiments about what could happen if portals were real!]

# 8-year-old Kid v. 45-year-old Pseudo AI

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I’ve variously been on again off again regarding computer/video games. Early on I had this fantasy concept that Leo would only be allowed to play computer games that he programmed himself, and that held for a little while, during which time we created some interesting, although laughably poor, games on the HopScotch platform. Since then we’ve engaged in several rounds of brief run-ins with puzzle games such as Where’s My Water and Monument Valley, and a couple others. We finally came to a mutual landing on Minecraft, which is at least slightly creative, and has a lot of programming and math_ed potential. Still, his media (including games) time is quite limited.

Another game that Leo has become enamored with (within the tight content and time controls that we put on him) is Portal, which is an extremely clever puzzle game with amazing graphics. (It has a tiny bit of robot violence, but is generally just a clever puzzle.) One of the best parts of Portal is that your adventure through the “lab” is narrated by an AI companion, named Glados (in Portal 2 it’s Wheatley). There some kind of complex back story, where Glados was a person uploaded into a computer, but Wheatley is a fully self-conscious AI. Of course, what these “AI”s say is completely canned, but it’s also (sometimes) clever and (often) funny.

Somewhat to my surprise, Leo’s interest in the games, combined with the (sometimes/often) clever/funny commentaries from the companion (pseudo) “AI”s has gotten Leo interested in AI more generally. Yesterday, Leo was monkeying with the Chegg flashcard app on my phone (for no reason other than wasting time on a long car ride), and, unbidden, he created some AI flashcards:

I have a bit of personal history with this sort of AI, having been the author (many many year ago — like, 1973!) of a BASIC version of Eliza that ran in very early PCs, and so became extremely popular. Indeed, I’ll bet that my Eliza is about the most knocked-off (as in copied/modified) program on the planet!

I happen to have a copy of Peter Schorn’s iAltair on my iPhone that coincidentally comes with a close knock-off of my actual old BASIC Eliza! So, since Leo is interested, having done the AI flash cards, I put him on my Eliza, now, um, 2018-1973=45 years later!

Here’s their conversation:

A while back we did a little toe-in-the-water experiment, using Lisp to write (bad) poetry. So, next week we’re going to start writing our own AI!

# Car Physics Revisted

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Long long time ago (actually, only back in 2015) we tracked a short road trip and manually computed and plotted the Time/Distance/Velocity/Acceleration. I’ve been wanting to revisit that fun little experiment, and this past weekend we had the opportunity on a 60-mile trip. Leo recorded our distance from start every five miles. This time, instead of manually plotting (which he does a lot of in school), I started him on Excel, which is certainly how he’ll be doing this once he gets beyond 3rd…or maybe 5th grade!

Here’re the results (you can see the master by clicking the pic):

The valley in the middle of the trip is where we stopped for lunch and turned around. And, yes, we really did travel exactly 60 miles, although that wasn’t planned!

# Calculus Craft

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Okay, so I’m a little slow! I’ve been working my way toward calculus with Leo for a long time. We’ve explored a great deal among the preliminaries, such as algebra, geometry, and trig, done a little “real life calculus” using cars and rockets, and I’ve even left various fun “calculus coloring books” around, which he’s picked up and variously paged through. We’ve even done some Minecraft-math before that verges on calculus. But until today I hadn’t really broached any real calculus.

Today it hit me that Minecraft provides a perfect setting to discuss integration via Riemann Sums (which most of you will remember as the method of rectangles). The reason is obvious: In Minecraft you (generally) only have rectangles, so if you want to know the area under something — say the roof of semi-circular building — you don’t have any choice but to build the thing out of blocks. Therefore, the actual area is actually most conveniently measured in terms of piles of blocks, i.e., rectangles, i.e., the basis of the Reimann Sums!

Once I realized this it took only a couple of minutes to come up with some easy examples. To you and me, this just looks like simple intro calculus, but the “patter” — the story I was telling all the way through — is all about Minecraft!

Next I explored the idea that if the blocks were smaller, the approximation would get better and better. Conveniently, the Minecraft blocks are 16-units wide, so you can divide them in half four times and still be working with integers, making it easy to actually work the detailed math.

What’s most interesting about this is that because of Leo’s facility with Minecraft, once he had his Minecraft thinking cap on (which, once on, is extremely hard to get off!), he was able to see right away how the simple version of the rectangle-based integration worked, and was also able to easily think through (approximately) how things would go if the blocks were divided in half, and then half again and again and again!

In the last example (second half of the page) Leo worked the Integral[X^2] by rectangles while I worked it by the usual (X^(N+1))/(N+1) method. (I had to help him a little with his.) And we both got about 42, which is about right! (Actually, it’s 125/3 = 41.666…)

I think that this (somewhat “duh”) realization of the relationship between Minecraft and Calculus has opened up our whole next level of math fun!