Problems & Puzzles: Puzzles

Puzzle 176. Primes in a circle*

Here we will ask you to distribute the first n integers (from 1 to n) around a circle such that:

• Every integer is used only once
• The sum of every two contiguous numbers is a prime
• The sum of every two opposite numbers is a prime

Soon you'll discover that solutions are possible only if n=even & n is not divided by 4

Example for n=10:

4 9
7        2
6            1
5       10
8  3

Question:

1.  Find the strategy in order to make that kind of assignations.

2. Find a solution for n=202

3. Explain why there is not any solution for n=6.

________
The question 1 of this puzzle was posed to my by a student from Brazil. At the moment I have discovered one strategy for making this task, and the calculation capability of my software has allowed to me to get solutions as large as n=750.

Solution:

Jud McCranie, Rudolf Knjzek and Phil Carmody solved the question 3. The shortest argument came Jud:

Each number has to have three other numbers that sum to primes - the two next to it and the one opposite it.  For n=6, 3 has only two (2 and 4).

***

Nobody has found/sent any strategy

***

But Jud found by found by exhaustive combinatorial analysis a solution for n=202:

1 2 3 4 7 6 5 8 9 10 13 16 15 14 17 12 11 18 19 22 21 20 23
24 29 30 31 28 25 34 27 26 33 38 35 32 39 40 43 36 37 42 41
48 49 52 45 44 53 50 47 54 55 46 51 56 57 70 61 66 65 62 69
58 73 64 63 68 59 72 67 60 71 78 79 84 83 74 75 76 81 82 85
88 91 90 77 80 87 86 93 98 95 96 97 94 99 92 89 102 109

130 111 100 123 106 121 108 119 104 129 154 115 112 117 110 101
126 113 120 151 118 153 116 107 122 149 132 139 124 103 136 105
134 135 128 141 140 171 196 157 114 125 138 131 150 127 166 147
146 161 152 179 168 145 148 183 184 169 162 185 164 189 160 133
178 159 194 143 174 199 202 181 186 163 190 193 156 155 158 201
188 195 142 175 192 167 200 197 170 177 176 173 144 137 180 187
172 165 182 191 198

***

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