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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## April 2022 - Solution

April 2022 Challenge

April 2022 Solution:A non-exact solution can be found using optimization algorithms, but it is possible to find a closed-form formula for the optimal solutions. An excellent description, given below, was sent to us by Anders Kaseorg -thanks Anders!

For all i + j + 1 = n, one optimal solution is

x = [
√(nj/i) − 1 repeated i times,
−√(ni/j) − 1 repeated j times,
n − 1
] / √(2n(n − 1)),
y = [
√(nj/i) + 1 repeated i times,
−√(ni/j) + 1 repeated j times,
−n + 1
] / √(2n(n − 1)).

For example, at i = 4, j = 15, n = 20, we get

x = [
0.277866618713, 0.277866618713, 0.277866618713, 0.277866618713,
-0.120044594164, -0.120044594164, -0.120044594164,
-0.120044594164, -0.120044594164, -0.120044594164,
-0.120044594164, -0.120044594164, -0.120044594164,
-0.120044594164, -0.120044594164, -0.120044594164,
-0.120044594164, -0.120044594164, -0.120044594164,
0.689202437605
]
y = [
0.350414243724, 0.350414243724, 0.350414243724, 0.350414243724,
-0.047496969153, -0.047496969153, -0.047496969153,
-0.047496969153, -0.047496969153, -0.047496969153,
-0.047496969153, -0.047496969153, -0.047496969153,
-0.047496969153, -0.047496969153, -0.047496969153,
-0.047496969153, -0.047496969153, -0.047496969153,
-0.689202437605
]

Some of these solutions are rational. An infinite family of them can be constructed using the Pell numbers

P[0] = 0,
P[1] = 1,
P[k] = 2P[k−1] + P[k−2],

as follows: for all k ≥ 0, one rational optional solution is

x = [
(P[2k+1] + P[2k+2] − 1)/P[4k+4] repeated P[2k+2]² times,
(−P[2k+1] − P[2k+2] − 1)/P[4k+4] repeated P[2k+2]² times,
(P[2k+2]² − 1)/P[4k+4]
],
y = [ (P[2k+1] + P[2k+2] + 1)/P[4k+4] repeated P[2k+2]² times,
(−P[2k+1] − P[2k+2] + 1)/P[4k+4] repeated P[2k+2]² times,
(−P[2k+2]² + 1)/P[4k+4]
].

For example, at k = 1, we get this solution with n = 289:

x = [
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51, 2/51,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
-3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68, -3/68,
12/17
]
y = [
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68, 3/68,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51, -2/51,
-12/17
]