Transcript Why Does the Reduction Work?

Using Reduction for the
Game of NIM
Nim
At each turn, a player chooses one pile and removes some
sticks.
The player who takes the last stick wins.
Problem: Is there a move that guarantees a win for the current
player?
Nim
● Obvious approach: search the
space of possible moves.
● Reduction to an XOR computation
problem:
100
101
010
011
1
1
0
10
01
11
Nim
● Obvious approach: search the
space of possible moves.
● Reduction to an XOR computation
problem:
100
101
010
011
1
1
0
10
01
11
● XOR them together:
♦ 0+ means state is losing for current player
♦ otherwise current player can win by making
a move that makes the XOR 0.
Correctness Proof of Nim Reduction
• If all 0’s, there is no move that preserves all 0’s:
• If not all 0’s, there is a move that makes it all 0’s:
Correctness Proof of Nim Reduction
• If all 0’s, there is no move that preserves all 0’s:
Suppose there were such a move. Then it would
change some row n.
Since it must subtract at least 1 from the number
in row n, it would necessarily change a 1 in
some column j to a 0.
11
Since the XOR of that column was 0, it now
becomes 1.
01
The only way to return it to 0 would be to alter
some other row in column j.
But a given move can only alter a single row. So
the new XOR will contain at least one 1.
10
00
Correctness Proof of Nim Reduction
• If not all 0’s, there is a move that makes it all 0’s:
1. Scan from the left to find the first column, j,
that contains a 1 in the XOR row.
2. Choose a row, i, that contains a 1 in column j.
Change the 1 to 0.
3. Continue moving from the left. If the XOR row
contains a 1 in column j, flip bit (i, j).
101
111
001
011
• At the end, the XOR row is all 0’s because we flipped exactly
one bit in every column that had been 1 and no bits in columns
that had been 0.
• The flipping corresponds to a legal move because it changes
only one row and it creates a row that represents a smaller
number than the original one.