Palindromes on a chessboard
Take a standard chessboard, and number the black squares 1 and the red squares 0, as shown.
How many palindromes does this board contain in 'word-search' fashion?
Details and Assumptions:
- A palindrome and its reversal should not be counted twice.
- Each instance of a number should be counted separately.
- A palindrome can be one-digit long, but zero and numbers with leading zeros are not allowed.
- Numbers can be read along any single line vertically, horizontally, or diagonally.
The answer is 268.
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1 solution
6 ∗ 1 6 + 2 ( 1 + 3 + 6 + 1 0 + 1 5 + 2 1 ) + 2 8 + 3 2 = 2 6 8
Don't count the single 1's until the very end to avoid overcounting them.
Every horizontal row and every vertical column has has 6: three 101, two 10101, one 1010101: 6 ∗ 1 6 = 9 6
Diagonals with zeros have none
Diagonals with n 1's have n(n-1)/2. For example 1111 has 6: three 11, two 111, one 1111.
There are two diagonals each of length 2 to 7: 2 ( 1 + 3 + 6 + 1 0 + 1 5 + 2 1 ) = 1 1 2
and a single diagonal of length 8: 2 8
Finally add in the 32 single 1's: 3 2
9 6 + 1 1 2 + 2 8 + 3 2 = 2 6 8