6
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Place seven different positive integers on the empty disks of the H figure below so that the product of the three numbers in any straight black line is always the same. Now place seven other numbers in the disks of a similar H so that the products are again equal, but precisely one more than on the first case.

What is the least those products can be?

enter image description here

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  • $\begingroup$ Do the seven other numbers have to be entirely different from the first set? Like if you had 1,2,3,4,5,6,7 in the first set and 2,3,4,5,6,7,8 would that be allowed? $\endgroup$ – gabbo1092 Nov 22 '18 at 17:11
  • $\begingroup$ @gabbo1092: Inevitably they will be otherwise products will not be consecutive. $\endgroup$ – Bernardo Recamán Santos Nov 22 '18 at 17:13
  • $\begingroup$ @BernardoRecamánSantos unless they share a 1. $\endgroup$ – hexomino Nov 22 '18 at 17:22
  • $\begingroup$ @hexomino: True, so let's say they don´t share a 1. $\endgroup$ – Bernardo Recamán Santos Nov 22 '18 at 17:24
  • $\begingroup$ Does "in any straight line" include the diagonals, or just along the black lines? $\endgroup$ – shoover Nov 22 '18 at 18:49
4
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By brute force:

First H with product 225:

  9     3
  1 45  5
 25    15
and second H with product 224:
  1     7
 14  2  8
 16     4
or
  1     8
 16  2  7
 14     4

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  • $\begingroup$ Three Magic H's with consecutive products can be construct using numbers 1274, 1275, 1276. Their existence is related to Ramanujan's highly composite numbers. $\endgroup$ – Bernardo Recamán Santos Nov 23 '18 at 18:19
1
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Here are consecutive Magic H´s of lengths 2, 3, ..., 8:

224 ((2, 7, 16), (1, 4, 56), (1, 2, 112))

225 ((3, 5, 15), (1, 9, 25), (1, 3, 75))


1274 ((2, 13, 49), (1, 7, 182), (1, 2, 637))

1275 ((3, 17, 25), (1, 5, 255), (1, 3, 425))

1276 ((2, 11, 58), (1, 4, 319), (1, 2, 638))


9162 ((2, 9, 509), (1, 3, 3054), (1, 2, 4581))

9163 ((7, 17, 77), (1, 11, 833), (1, 7, 1309))

9164 ((2, 29, 158), (1, 4, 2291), (1, 2, 4582))

9165 ((3, 13, 235), (1, 5, 1833), (1, 3, 3055))


17574 ((2, 29, 303), (1, 3, 5858), (1, 2, 8787))

17575 ((5, 37, 95), (1, 19, 925), (1, 5, 3515))

17576 ((2, 13, 676), (1, 4, 4394), (1, 2, 8788))

17577 ((3, 9, 651), (1, 7, 2511), (1, 3, 5859))

17578 ((2, 17, 517), (1, 11, 1598), (1, 2, 8789))


63423 ((3, 27, 783), (1, 9, 7047), (1, 3, 21141))

63424 ((2, 8, 3964), (1, 4, 15856), (1, 2, 31712))

63425 ((5, 43, 295), (1, 25, 2537), (1, 5, 12685))

63426 ((2, 11, 2883), (1, 3, 21142), (1, 2, 31713))

63427 ((7, 17, 533), (1, 13, 4879), (1, 7, 9061))

63428 ((2, 101, 314), (1, 4, 15857), (1, 2, 31714))


179330 ((2, 79, 1135), (1, 5, 35866), (1, 2, 89665))

179331 ((3, 113, 529), (1, 23, 7797), (1, 3, 59777))

179332 ((2, 107, 838), (1, 4, 44833), (1, 2, 89666))

179333 ((7, 17, 1507), (1, 11, 16303), (1, 7, 25619))

179334 ((2, 9, 9963), (1, 3, 59778), (1, 2, 89667))

179335 ((5, 31, 1157), (1, 13, 13795), (1, 5, 35867))

179336 ((2, 29, 3092), (1, 4, 44834), (1, 2, 89668))


294590 ((2, 89, 1655), (1, 5, 58918), (1, 2, 147295))

294591 ((3, 79, 1243), (1, 11, 26781), (1, 3, 98197))

294592 ((2, 8, 18412), (1, 4, 73648), (1, 2, 147296))

294593 ((13, 31, 731), (1, 17, 17329), (1, 13, 22661))

294594 ((2, 37, 3981), (1, 3, 98198), (1, 2, 147297))

294595 ((5, 19, 3101), (1, 7, 42085), (1, 5, 58919))

294596 ((2, 47, 3134), (1, 4, 73649), (1, 2, 147298))

294597 ((3, 27, 3637), (1, 9, 32733), (1, 3, 98199))

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