# What do you think of this educational game?

It is not easy to think of games for young children that are both educational and strategically interesting. See what you think of this.

The game is for two players, who alternate turns. Start with the numbers 2 through 10. Each player in turn gets to remove a single number and all numbers divisible by it. The winner is the one who takes the last number. The game can be played with pencil and paper.

If I were more ambitious, I would write a computer program for the game, which would allow either two players or one person playing against the computer. I did write a program to find out how to win the game. If you want to try this on your own, with or without a computer, I provide the hint that the game is a win for the first player and there is only one winning first move. I actually guessed the winning move, but used the computer to work out all the different cases.

If you are mathematically inclined and interested in a readable account of how to determine winning moves for games like this, I learned about the theory from this online paper, which is part of a larger book. I used the section on N and P positions, which is how far I have read so far.

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## 7 Answers

It sounds like an educational game, but as a kid I most likely would not have thought it was very fun. It’s pretty difficult to create a game that is both educational and fun, especially for the majority of kids. It’s a fine line, it is often educational but not fun, or vice versa. I’m sure some kids would find this game fun, but I’m not sure if the majority of kids would.

I might have enjoyed playing it when I was bored as a child, but as @mangeons said, it wouldn’t be appealing to everyone. I vastly preferred (and still do) games or puzzles involving words rather than numbers.

I don’t think that it makes a fun game. It is an abstract strategy game, which means that it has a game tree. If you are trying to make an abstract strategy game, then the game must have a few things to make it fun. It must be very hard to pick a “best” move, but it must be easy to see a “good” move. It must also be possible to tell who is winning.

In its current state, I feel like the game involves too much guesswork. At a small scale, the game tree is too easy to memorize, but on a large scale, the game tree becomes chaotic. It is impossible to intuitively tell which moves are better than others, and it is hard to tell who is winning until the game is almost over.

Okay, I get the points. When I thought of it, it seemed kind of interesting, but I was not sure.

@PhiNotPi makes several good points as to what characteristics make a game interesting. I can see that the game I came up with is too abstract. It does help to have some sort of story or metaphor to wrap a game around. I also agree that it should also be possible to distinguish good moves from bad ones. As to whether an interesting game requires being able to tell who is ahead, I am not sure. Certainly that applies in athletic contests. Can you always tell who is ahead in a game of chess? One thing that chess does allow for is global strategies, which my game definitely does not.

Hmm. At first it reminded me of nim, which has a known strategy that kids catch on to. But your game involves eliminating *multiples* of chosen numbers, which reminds me more of the sieve of Eratosthenes for finding prime numbers. Educationally it’s wonderful, but the unpredictable distribution of primes & composites might make this game intractably difficult to analyze for larger numbers, including the “backward induction” method in @LostInParadise‘s OP link.

Each turn one or more numbers are eliminated, leaving a survivor set. For example if player A goes first & chooses 2, then all even numbers are eliminated, leaving {3,5,7,9}. Sooner or later the survivor set gets reduced to a set of elements none of which is a multiple of any other. Once a player reaches this condition, each successive turn eliminates just one element, so who wins is determined by whether the number of elements is even or odd when this happens.

Even with that in mind I tried to work out a game tree to find a winning strategy & got bogged down in details more quickly than I expected.

Make the numbers up to 100, say, and I think it’s already very difficult except by computer. Or am I missing something clever?

There does not seem to be a general strategy for the game. 10 seems to be about the best compromise between being trivial and being what @PhiNotPi describes as “chaotic.”

For what it is worth, the winning move is to take 7.

Here are winning responses to all the possible moves by the second player, at which point the rest of the game should be fairly easy to analyze. Note that, except for one case, the move and response are interchangeable, leaving only 5 cases to analyze.

2,4,6,8,10 – 9

3,6,9 – 8

4,8 – 5,10

5,10 – 4,8

6 – 9

8 – 3,6,9

9 – 2,4,6,8,10

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