Nim and simple mathematical proofs 2/3

Nimble is a game where there is a row of squares numbered 0, 1, 2, 3, … Each square can contain a coin, and a square may contain multiple coins. The game is played by two players, taking turns in moving a single coin to a lower-numbered square. A player must move and exactly one coin during his turn. A coin may only be mover to a lower-numbered square. The game is won by the player who moves the last coin into square 0. At first this game might look like a distant cousin of Nim, but actually this game is exactly Nim, just in disguise. Let’s see the “proof” for that.

I will present below the rules of Nim and the rules of Nimble to show that the games are identical. Each rule contains the analogous version of the rule for both games.

Rule 1:
Nim: Two players take turns in removing sticks from piles. In the beginning, there are at least three piles.
Nimble: Two players take turns in moving one coin from a square to a lower-numbered square.

Rule 2:
Nim: During his turn a player must remove at least one stick from one pile.
Nimble: During his turn a player must move one coin to at least the next lower-numbered square.

Rule 3:
Nim: A player may remove sticks from one pile only during one turn.
Nimble: A player may move only one coing during one turn.

Rule 4:
Nim: A player to remove the last stick wins
Nimble: A player to move the last coin to square 0 wins.

Rule 5:
Nim: The piles do not interact with each other.
Nimble: The squares and coins do not interact with each other.

From rules 1–5 it is easy to see that Nim and Nimble are the same game with slightly different representations. Especially from rules 2 and 3 we can observe that removing of k sticks from exactly one pile of n sticks and moving exactly one coin from square n to square k are equivalent.

Although this “proof” is hardly very deep-going, or even a real proof for that matter, it shows well how one concept can be presented in different ways and how seemingly different situations can be just different representations of the same idea. This observation is a good reminder for everyday life. When facing novel problems (e.g. Nimble), we should ask ourselves if we could draw parallels to known and familiar problems (e.g. Nim). On the other hand, when looking at existing ideas and concepts (e.g. Nim) we might want to try formulating them a bit differently (e.g. Nimble) to see if we can gain more insight into them.

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