perplexus.info :: Just Math : 1 Palindrome 1

1 Palindrome 1 (Posted on 2006-06-10)

Take a standard 8x8 chessboard, number the black squares 1 and the white squares 0. How many palindromes does this board contain in 'word-search' fashion?

Now generalize and find a formula for any n*n board.

Notes: A palindrome and its reversal should not be counted twice. A palindrome must be at least two digits long and leading zeroes are not allowed. Numbers can be read in any single vertical, horizontal or diagonal direction.

Submitted by Jer

Rating: 3.6667 (3 votes)

Solution: (Hide)

For an 8x8 the answer is 6*16+2(1+3+6+10+15+21) + 28 = 236
The general formula for even n is:
(7/12)n³ - n² + (1/6)n
For odd n and black corners:
n³ -8n² + 15n + 4 + 4[sum of odd triangular numbers up to n-2]
For odd n and white corners:
n³ -8n² + 17n + 4 + 4[sum of even triangular numbers up to n-3] + 2[n-1^st triangular number]
Note: the two odd formulas add to twice the even formula

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-04-03 12:17:41
some numbers by actual count	Charlie	2006-06-11 10:08:42
re: partial solution	Charlie	2006-06-11 09:52:16
Triangular numbers	brianjn	2006-06-11 08:17:22
re: question	Justin	2006-06-10 18:44:54
question	Charlie	2006-06-10 18:34:59
partial solution	Justin	2006-06-10 17:30:17
re: Some numbers	Phil_Osopher	2006-06-10 17:29:14
Some numbers	e.g.	2006-06-10 16:24:54

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