The elder Free-Ride offspring has lately gotten into playing “poison”, a nim-type game for two players. You start with a pile of twelve items that are the same and one item that is different (the poison). Each turn, players can remove either one or two items from the pile. The object of the game is to leave your opponent with no option but to take the poison.
In theory, it is possible to win the game every single time if your turn is second. (Thanks to MarkP for straightening me out on this one.) What the elder Free-Ride offspring has discovered in playing with the younger Free-Ride offspring, however, is that there are circumstances in which a five-year-old will find it psychologically impossible to exercise a winning strategy even when given the second turn.
Say, for example, the twelve items that are the same are M&Ms. Even though it may be advantageous (from the point of view of sticking your opponent with the poison) to take only one M&M from the pile when it’s your turn, a five-year-old will always take two M&Ms from the pile. More M&Ms are always better than fewer, aren’t they?
Similarly, if the poison is some very attractive item (like a gingerbread person), a five-year-old who is already hell-bent on maximizing the M&M take will not feel terribly put out at being stuck with the poison. You’ve lost? How is it, then, that you’ve scored six M&Ms and a cookie?
It remains to be seen whether playing the game with dried beans and a turnip dreate conditions in which a five-year-old can develop a winning strategy.
It sounds like a good game for teaching the value of questioning one’s assumptions, even if a 5-year old might not see it that way.
Speaking of which, my take on the game is that you can win every time if you go second, by always choosing the opposite amount your opponent chooses, thus removing three each round and leaving him with the poison to choose to begin round 5. Am I misreading something?
MarkP, you’re absolutely right that the second turn is the one from which it is always possible to win. (As I was mostly a nearby observer of the poison games, I got mixed up.)
SPOILER! SPOILER! SPOILER! SPOILER! SPOILER! Full explanation of the strategy below! Don’t look at it until you’ve played a few games of poison and worked out your prefered strategy on your own!
I want you to be stuck with the poison and no M&Ms to take. (If there’s even one M&M, you can take that M&M and avoid taking the poison.)
If I can leave 3, you will lose (you only can take 1 or 2, not all three)
So, you don’t want to leave me 4 or 5 (because then I can leave you 3)
But, if I leave you 6, you have to leave me 4 or 5.
So, you don’t want to leave me 7 or 8 (because then I’ll be able to leave you 6)
If I can leave you 9, you have no choice but to leave me 7 or 8.
So, you don’t want to leave me 10 or 11 (which would let me give you 9).
But, if you go first (where there are 12 M&Ms) you have no choice but to leave me 10 or 11.
So, going second, it should always be possible to win — providing you can remember the strategy!
Leave 9, then leave 6, then leave 3, then take the remaining M&M(s).
And, while we’re on the subject of poison, you might enjoy Chomp, a don’t-eat-the-poison-cookie game that maps onto Nim.
At that age (5-6), my approach to games involving strategy was to re-write the agenda. I’d throw out the supposed goal, pick my own goal, and pursue that. It had some pretty entertaining effects on the other players too.
A dozen years later, I discovered the same strategy let me reliably beat some faily seasoned chess players — they would always be convinced I was following some subtly calculated strategy, when I was really just playing to confuse my opponent. Once the opponent was sufficiently bewildered and the board cleared of more than half the pieces, swooping in for the kill was a lot easier. Scarily enough, this would still work after I told them what I was doing.
Before I went off to college, aged 16, I used to do a lot of babysitting. I’d teach the kids poker, as part of the parental agreement that I’d be teaching them Math. I’d win some of their allowances. That would motivate them powerfully to learn to play better.
I did, however, lose one game of 20 Questions badly. First I lost a round where the brother and sister were thinking of a vegetable that began with the letter “P.” Namely: “Pusghetti.”
I lost the next round. I established early that they were thinking of something neither solid, nor liquid, nor gas. After that, every guess I made about plasmas, electricity, abstractions, convulsed them with laughter.
Having exhausted all 20 questions, I gave up. “It’s right in front of your eyes,” they hinted. Then one of them took off my eyeglasses.
“But those are solid!” I said, outraged.
“Don’t be silly,” said the girl. And she folded the earpieces along the hinges. “Not solid at all.”
OK, that’s the 3n version, how about the 3n+1 and 3n+2 versions?
(cryptic hint)
Bob
This is really very interesting – preschool aged children have difficulty controlling “prepotent” responses. Generally this involves anything with which they have previous experience, or anything which is intrinsically rewarding. Stephanie Carlson and colleagues have some really interesting work on the “less is more” game, in which you must select a smaller amount of items in order to win the larger amount. Kids are really bad at this when it involves something like M&M’s, but magically improve if you use something more abstract, like dots or stones.
Pretty neat… thanks!
It’s not just five year olds affected by this. I remember at college once playing chess with those chessmen cookies that Pepperidge Farm used to produce. The deal was that you got to eat what you captured, and whatever was left on the board if you won. (Though in practice that last bit would be split)
Anyway, over the course of three games (which used up our supply of pieces) it became obvious that this changed the chess strategy to a very bloody game, with lots of even trades happening.