Cliff Pickover never fails to be disappointingly facile. Shiver in ecstasy indeed.
The first comment in that Twitter thread is much better. There are an infinite sequence of these things. Suppose we call the largest term on the left x², then we are solving (x-k)² ... + (x-1)² + x² = (x+1)² + (x+2)² ... (x+k)² The equations balance when x is twice the sum of the x-term coefficients. i.e. x is four times a triangular number. For three terms: 3² + 4² = 5² (4 = 4 * 1) four terms: 10² + 11² + 12² = 13² + 14² (12 = 4 * 3) five terms as above (24 = 4 * 6) etc
If you multiply all those (x-1)^2 terms on the left side of the equation and add them up, twice the multiplying term on the x will be the magic number. Next two numbers are 40 and 60
Cliff Pickover never fails to be disappointingly facile. Shiver in ecstasy indeed.
ReplyDeleteThe first comment in that Twitter thread is much better. There are an infinite sequence of these things.
Suppose we call the largest term on the left x², then we are solving
(x-k)² ... + (x-1)² + x² = (x+1)² + (x+2)² ... (x+k)²
The equations balance when x is twice the sum of the x-term coefficients. i.e. x is four times a triangular number.
For three terms: 3² + 4² = 5² (4 = 4 * 1)
four terms: 10² + 11² + 12² = 13² + 14² (12 = 4 * 3)
five terms as above (24 = 4 * 6)
etc
enlightening > mysterious
I love this. Thank you for commenting! While the equation is interesting, seeing the mechanics underlying it is so powerful!
DeleteIf you multiply all those (x-1)^2 terms on the left side of the equation and add them up, twice the multiplying term on the x will be the magic number. Next two numbers are 40 and 60
ReplyDelete