Subtract a square

Subtract-a-square (also referred to as take-a-square) is a two-player mathematical game of strategy starting with a positive integer and both players taking turns subtracting a non-zero square number not larger than the current value. The game is usually played as a normal play game, which means that the last person who can make a subtraction wins.

Illustration

A normal play game starting with the number 13 is a win for the first player provided he does start with a subtraction of 1:

player 1: 13 - 1*1 = 12

Player 2 now has three choices: subtract 1, 4 or 9. In each of these cases, player 1 can ensure that within a few moves the number 2 gets passed on to player 2:

player 2: 12 - 1*1 = 11        player 2:       12 - 2*2 = 8               player 2: 12 - 3*3 = 3
player 1: 11 - 3*3 =  2        player 1:        8 - 1*1 = 7               player 1:  3 - 1*1 = 2
                               player 2:  7 - 1*1 = 6 or: 7 - 2*2 = 3   
                               player 1:  6 - 2*2 = 2     3 - 1*1 = 2

Now player 2 has to subtract 1, and player 1 subsequently does the same:

player 2:  2 - 1*1 = 1
player 1:  1 - 1*1 = 0
  player 2 loses

Mathematical theory

In the above example, the number '13' represents a winning or 'hot' position, whilst the number '2' represents a losing or 'cold' position. Given an integer list with each integer labeled 'hot' or 'cold', the strategy of the game is simple: try to pass on a 'cold' number to your opponent. This is always possible provided you are being presented a 'hot' number. Which numbers are 'hot' and which numbers are 'cold' can be determined recursively:

1) the number 0 is 'cold', whilst 1 is 'hot'

2) if all numbers 1 .. N have been classified as either 'hot' or 'cold', then
  2a) the number N+1 is 'cold' if only 'hot' numbers can be reached by subtracting a positive square
  2b) the number N+1 is 'hot' if at least one 'cold' number can be reached by subtracting a positive square

Using this algorithm, a list of cold numbers is easily derived:

0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, 44, … (sequence A030193 in OEIS)

Cold numbers tend to end in 0, 2, 4, 5, 7, or 9. Cold values that end with other digits are quite uncommon. This holds in particular for cold numbers ending in 6. Out of all the over 180,000 cold numbers less than 40 million, only one ends in a 6: 11,356.^[1]

Extensions

The game subtract-a-square can also be played with multiple numbers. At each turn the player to make a move first selects one of the numbers, and then subtracts a square from it. Such a 'sum of normal games' can be analysed using the Sprague–Grundy theory. This requires the positions in the game subtract-a-square to be mapped onto equivalent nim heap sizes; the first few values are:

0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 2, 3, 4, … (sequence A014586 in OEIS).

Notice that all 'cold' positions get mapped onto a zero heap size.

Misère game

Subtract-a-square can also be played as a misère game, in which the player to make the last subtraction loses. The recursive algorithm to determine 'hot' and 'cold' numbers for the misère game is the same as that for the normal game, except that for the misère game the number 1 is 'cold' whilst 2 is 'hot'. It follows that the cold numbers for the misère variant are the cold numbers for the normal game shifted by 1:

Misère play 'cold' numbers:
1, 3, 6, 8, 11, 13, 16, 18, 21, 23, 35, 40, 45, ...

References