Un Max de singes

 

Entrée de l’atelier de Gabriel Max, ~1885, Richard Wintzer

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Square my chunks and add

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The idea is to transform the integer a into b and iterate. Cut a into the number of chunks you want, square them, add those squares. Example a = 1023 can be transformed in b  (last column) like this: 1 ² + 0 ² + 2 ² + 3 ² = 1 + 0 + 4 + 9  = 14 = b 1 ² + 0 ² + 23 ²     = 1 + 0 + 529    = 530   10 ² + 2 ² + 3 ²     = 100 + 4 + 9    = 113   10 ² + 23 ²         = 100 + 529      = 629   102 ² + 3 ²         = 10404 + 9      = 10413   1023 ²                              = 1046529   Question #1 Starting with 2 and iterating, what is the shortest path ending in 1? Here are 11 steps (left column) – can you do better? 2    –> 2² = 4 4    –> 4² = 16 16   –> 1² + 6² = 1 + 36 = 37 37   –> 37² = 1369...
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Triples for the new year

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(Dall-e creation ) The hereunder sequence  S  is both a succession of triples of monotonically increasing numbers and a succession of triples of monotonically increasing digits. S  is the lexicographically earliest such sequence with distinct number-triples: S = 0,1,2,0,1,3,0,1,4,0,1,5,0,1,6,0,1,7,0,1,8,0,1,9,0,2,3,0,2,4,0,2,5,0,2,6,0,2,7,0,2,8,0,2,9,0,3,4,0,3,5,0,3,6,0,3,7,0,3,8,0,3,9,0,4,5,0,4,6,0,4,7,0,4,8,0,4,9,0,5,6,0,5,7,0,5,8,0,5,9,0,6,7,0,6,8,0,6,9,0,7,8,0,7,9,0,8,9,1,2,3,1,2,4,1,2,5,1,2,6,1,2,7,1,2,8,1,2,9,1,3,4,1,3,5,1,3,6,1,3,7,1,3,8,1,3,9,1,4,5,1,4,6,1,4,7,1,4,8,1,4,9,1,5,6,1,5,7,1,5,8,1,5,9,1,6,7,1,6,8,1,6,9,1,7,8,1,7,9,1,8,9,2,3,4,2,3,5,2,3,6,2,3,7,2,3,8,2,3,9,2,4,5,2,4,6,2,4,7,2,4,8,2,4,9,2,5,6,2,5,7,2,5,8,2,5,9,2,6,7,2,6,8,2,6,9,2,7,8,2,7,9,2,8,9,3,4,5,3,4,6,3,4,7,3,4,8,3,4,9,3,5,6,3,5,7,3,5,8,3,5,9,3,6,7,3,6,8,3,6,9,3,7,8,3,7,9,3,8,9,4,5,6,4,5,7,4,5,8,4,5,9,4,6,7,4,6,8,4,6,9,4,7,8,4,7,9,4,8,9,5,6,7,5,6,8,5,6,9,5,7,8,5,7,9,5,8,9,6,7,8,6,7,9,6,8,9,7,...
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Du recuit simulé

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Toujours trouvé que  cette expression  était magnifique ! Et voilà qu'elle devient (un peu) mienne grâce au triangle ci-dessous (grâce aussi à Jean-Charles M . et  Mark K .)  Tout a commencé par un message (raté) sur Math-Fun et sa correction : Hello Math-Fun, no cell of the hereunder triangle shares any of its digits with another cell of the same rank or column: +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |  10 |  20 |  30 |  40 |  50 |  60 |  70 |  80 |  90 |        +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |  22 |  11 | 777 | 666 | 888 | 333 | 444 |   9 |                +-----+-----+-----+-----+-----+-----+-----+-----+ |  33 |   8 | 555 | 111 | 222 |  47 |  69 |                +-----+-----+-----+-----+-----+-----+-----+ |  44 |  ...
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