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What number should go in the place of ? and why?

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1 Answer 1

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A suitable answer would be:

123 (but equally 127, 128 and even more...)

Because whilst at first this question looks like it revolves around...

...the numbers on the bottom diagonal being the sum of the two numbers whose arrows point to it...

...this logic fails for the final two spaces on the diagonal, since...

...$113 + 6$ would be $119$, but $119 + 1000$ does not equal $1123$.

So we are looking for a different property of these numbers. And if we think laterally we can demonstrate a rule revolving around...

...the number of letters in the names of these numbers when spelled out (in English).

As follows:

NINETYEIGHT = 6 letters in NINETY, 5 letters in EIGHT
NINETYEIGHT = 11 letters (6+5) in NINETY EIGHT

NINETYEIGHTNINE = 11 letters in NINETY EIGHT, 4 letters in NINE
ONEHUNDREDSEVEN = 15 letters (11+4) in ONE HUNDRED SEVEN

ONEHUNDREDSEVENSIX = 15 letters in ONE HUNDRED SEVEN, 3 letters in SIX
ONEHUNDREDTHIRTEEN = 18 letters (15+3) in ONE HUNDRED THIRTEEN

ONEHUNDREDTHIRTEENSIX = 18 letters in ONE HUNDRED THIRTEEN, 3 letters in SIX
????????????????????? = 21 letters (18+3) required in the answer

?????????????????????ONETHOUSAND = 21 letters in the required answer, 11 letters in ONE THOUSAND
ONETHOUSANDONEHUNDREDTWENTYTHREE = 32 letters (21+11) in ONE THOUSAND ONE HUNDRED TWENTY THREE

In other words...

...when you count the number of letters that make up the names of the circled numbers spelled out, there are the same number of letters in the names of the numbers on the bottom diagonal as there are in those of the two circle numbers whose arrows point to it.

Thus, the missing number should have 21 letters when spelled out. The next number above 113 with this property is 123 (ONE HUNDRED TWENTY THREE), which also pleasingly happens to work arithmetically for the last step in the false surface pattern, so let's choose that! Numbers like 127, 128, 133, etc. would all also work - they're just not quite as pleasing as choosing the one that keeps up the deception of the initial gut-reaction pattern guess...

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    $\begingroup$ Wonderfully explained @Stiv $\endgroup$
    – RogerA
    Jan 7 at 23:38

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