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What phrase is hidden below?


               3  ? 
               5    ??
               8       ??
               7          ??
               20            ??
               12               ??
               9                   ??
               28                     ??
               11                        ??
               16                           ??
               33                              ??
               48                                 ??
               13                                    ??
               36                                       ??  
               39                                          ??
               65                                             ??
                                     
                  4 12 15 24 21 35 40 45 60 63 56 55 84 77 80 72

enter image description here

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1 Answer 1

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The triangle represents

The primitive Pythagorean Triples! It acts as row and column, so for instance the first '?' is 5, as it is the first Pythagorean triple, $3^2 + 4^2 = 5^2$, and works down the diagonal in that manner:

enter image description here

So

Calculating the remaining number from the triples, we get:

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 65, 73, 85, 85, 89 and 97

So the largest number of the triples in ascending order - and from the title they are all less than 100 (In fact, they are the exact 16 triples that are all below 100).

This all makes sense and it is hinted by the '??'s being on the hypotenuse of a right angled triangle - where Pythagoras' theorem applies. Finally, comparing the numbers with the grid below gives us the phrase:

PRIMITIVE TRIPLES!

Which is very fitting!

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2
  • $\begingroup$ Woah solved already??! Nice job, I didn’t even know there was a whole list of them until today! Isn’t the layout also of a right angled triangle hinting at this you could say? $\endgroup$
    – PDT
    Commented May 8 at 18:18
  • $\begingroup$ @PDT definitely, quite a few hints in the format which is all very satisfying. Updated the answer to include that, good point! $\endgroup$ Commented May 8 at 18:26

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