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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, 2018 at 17:11
  • $\begingroup$ @gabbo1092: Inevitably they will be otherwise products will not be consecutive. $\endgroup$ Nov 22, 2018 at 17:13
  • $\begingroup$ @BernardoRecamánSantos unless they share a 1. $\endgroup$
    – hexomino
    Nov 22, 2018 at 17:22
  • $\begingroup$ Does "in any straight line" include the diagonals, or just along the black lines? $\endgroup$
    – shoover
    Nov 22, 2018 at 18:49
  • $\begingroup$ @hexomino: just along black lines. $\endgroup$ Nov 23, 2018 at 0:24

2 Answers 2

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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. $\endgroup$ Nov 23, 2018 at 18:19
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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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