Here are the rules I devised on the spot [1]: The frame contains 5 pegs of different height, each of which takes, 1, 2, 3, 4, or 5 balls. There are 15 balls in five colors, with 1, 2, 3, 4, or 5 balls in each color. The game starts with all of the balls off of the frame.
On your turn you must do ONE of the following: 1) Take as many balls as you like that are all the same color and put them all onto a single peg, or 2) Take exactly two balls of different colors and place them onto two different pegs.
The peg(s) must have room to accommodate the balls you selected. The peg(s) on which you place the balls can have other balls on it from any colors; you don't have to match the ball colors already on those pegs. When choosing the first option, you can take any number of balls you want of the same color; for instance, if there are four yellow balls, you can take 1, 2, 3, or all 4, as long as there is space on a a single peg to place them.
As in other nim games, the last player to play loses.
Analysis
There are only 15 balls, which makes this a very limited game. What complicates this is the varying color choices and limitations imposed by the spaces on the pegs.
Let's refer to the available balls at the start as (5,4,3,2,1) and the available frame spaces as [5,4,3,2,1]. As options disappear, we can ignore ball colors and spaces that have no available choices. So, for example, if there are three balls remaining in 2 colors, you can select (1), (2), or (1,1), but only if the frame spaces allow for these selections. If the remaining frame space is [3], you can't select (1,1). If the remaining frame space is [1,1,1], you can't select (2).
First Moves
For your first move, you have 25 choices for selection: 1 ball from (1), 1 ball from (2), 2 balls from (2), 1 ball from (3), etc up to 5 balls from (5): that's 15. You have 10 more choices for (1,1) from the 5 available colors. That's the selection.
Now you have to place the balls. If you chose (1) (5 different possibilities), you have 5 choices for placing. That's 25 possibilities.
(2) (4 choices) x 4 placement possibilities = 16
(3) (3 choices) x 3 placements = 9
(4) (2 choices) x 2 placements = 4
(5) (1 choice) x 1 placement = 1
(1,1) (10 choices) x 10 placements = 100
Add it all together and you get a total of 155 options for your first move.
End Games
Let's work backwards, instead.
1 ball
You lose if it is your turn and there is one ball.
2 balls
You win if it is your turn and there are two balls. Place one on the frame, and now your opponent has 1 ball.
3 balls
There are nine options for 3 balls. In some configurations you can force a win, and in others you have lost.
| State | Balls | Frame | Win/Loss | Comments |
|---|---|---|---|---|
| 3.1 | (3) | [3] | Win | Place (2), leaving your opponent with one ball. |
| 3.2 | (3) | [2,1] | Win | Place (2), leaving your opponent with one ball. |
| 3.3 | (3) | [1,1,1] | Loss | You must place (1), leaving your opponent with two balls. |
| 3.4 | (2,1) | [3] | Win | Place (2), leaving your opponent with one ball. |
| 3.5 | (2,1) | [2,1] | Win | Place (2) or (1,1), leaving your opponent with one ball. |
| 3.6 | (2,1) | [1,1,1] | Win | Place (1,1), leaving your opponent with one ball. |
| 3.7 | (1,1,1) | [3] | Loss | You must place (1), leaving your opponent with two balls. |
| 3.8 | (1,1,1) | [2,1] | Win | Place (1,1), leaving your opponent with one ball. |
| 3.9 | (1,1,1) | [1,1,1] | Win | Place (1,1), leaving your opponent with one ball. |
4 balls
There are twenty five options for 4 balls. In some configurations you can force a win, and in others you have lost.
| State | Balls | Frame | Win/Loss | Comments |
|---|---|---|---|---|
| 4.1 | (4) | [4] | Win | Place (3), leaving your opponent with one ball. |
| 4.2 | (4) | [3,1] | Win | Place (3), leaving your opponent with one ball. |
| 4.3 | (4) | [2,2] | Loss | If you place (2), you leave your opponent with two balls. If you place (1), your opponent is left in state 3.5. |
| 4.4 | (4) | [2,1,1] | Win | Place (1) in the [2], leaving your opponent in state 3.3. |
| 4.5 | (4) | [1,1,1,1] | Win | Place (1), leaving your opponent in state 3.3. |
| 4.6 | (3,1) | [4] | Win | Place (3), leaving your opponent with one ball. |
| 4.7 | (3,1) | [3,1] | Win | Place (3), leaving your opponent with one ball. |
| 4.8 | (3,1) | [2,2] | Loss | If you place (2) or (1,1), you leave your opponent with two balls. If you place (1), your opponent is left either in state 3.2 or 3.5. |
| 4.9 | (3,1) | [2,1,1] | Win | Place the single ball in [2], leaving your opponent in state 3.7. |
| 4.10 | (3,1) | [1,1,1,1] | Win | Place the single ball, leaving your opponent in state 3.7. |
| 4.11 | (2,2) | [4] | Loss | If you place (1) or (2), your opponent places (2) or (1) respectively. |
| 4.12 | (2,2) | [3,1] | Loss | If you place (1), your opponent places (2). If you place (2) or (1,1), your opponent is left with two balls. |
| 4.13 | (2,2) | [2,2] | Loss | If you place (1), your opponent places (2). If you place (2) or (1,1), your opponent is left with two balls. |
| 4.14 | (2,2) | [2,1,1] | Loss | If you place (1), your opponent places (1,1) or (2). If you place (2) or (1,1), your opponent is left with 2 balls. |
| 4.15 | (2,2) | [1,1,1,1] | Loss | If you place (1), your opponent places (1,1). If you place (1,1), your opponent is left with 2 balls. |
| 4.16 | (2,1,1) | [4] | Win | Place (1) from the (2), leaving your opponent in state 3.7. |
| 4.17 | (2,1,1) | [3,1] | Win | Place (1) from the (2) in the [1], leaving your opponent in state 3.7. |
| 4.18 | (2,1,1) | [2,2] | Loss | If you place (2) or (1,1), you leave your opponent with two balls. If you place (1), your opponent is left in state 3.5 or 3.8. |
| 4.19 | (2,1,1) | [2,1,1] | Loss | If you place (1), your opponent places (1,1). If you place (2) or (1,1), you leave your opponent with two balls. |
| 4.20 | (2,1,1) | [1,1,1,1] | Loss | If you place (1), your opponent places (1,1). If you place (1,1), you leave your opponent with two balls. |
| 4.21 | (1,1,1,1) | [4] | Win | Place (1), leaving your opponent in state 3.7. |
| 4.22 | (1,1,1,1) | [3,1] | Win | Place (1) in the single spot, leaving your opponent in state 3.7. |
| 4.23 | (1,1,1,1) | [2,2] | Loss | If you place (1), your opponent places (1,1). If you place (2) or (1,1), you leave your opponent with two balls. |
| 4.24 | (1,1,1,1) | [2,1,1] | Loss | If you place (1), your opponent places (1,1). If you place (2) or (1,1), you leave your opponent with two balls. |
| 4.25 | (1,1,1,1) | [1,1,1,1] | Loss | If you place (1) or (1,1), your opponent places (1,1) or (1) respectively. |
Note that you always lose if the frame configuration is [2,2] on your turn, and you always lose if you're left with (2,2) on your turn.
5 balls and more
There are 49 options for 5 balls. Things continue to grow, but slower, for more balls, since limitations on spaces and ball colors begin to apply. The whole thing is not terribly hard to work out with a computer, but it doesn't appear to be easy to figure out for humans, unless someone hands you the winning strategy.
Variants
I created the game after showing my daughter the Towers of Hanoi puzzle on this. It's not hard to add a number of twists to the game to change the challenge. For example, you could say that certain colors are not allowed to be placed next to or on top of (like Towers of Hanoi) other colors.
Yehuda
[1] I vaguely recall rules similar to these in other games I have played, so the idea isn't original, although I think the game is.
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