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Arrange numbers 1 to 40 to the table with rules:

  • Multiple of 4 Numbers and some perfect square numbers are already there as guide.
  • Every 3 consecutive numbers with pattern white-yellow-white, must sums to 50 or 100.
  • There is only 1 unique solution.
  • Some smart guessing are needed.


enter image description here

This is the sample for sums to 40 or 80.
enter image description here

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  • $\begingroup$ Do you mean that all multiples of 4 and all perfect squares are filled in, so none of the blanks will take those values? $\endgroup$ – Jasen Nov 21 '17 at 4:07
  • $\begingroup$ @jasen ok sory except 1. $\endgroup$ – Jamal Senjaya Nov 21 '17 at 4:16
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My solution (I didn't verify uniqueness):

Solution

Here's generally how I solved it:

The first step was to place $6$, $13$, and $37$ (and then $38$). Then I looked at the square between $20$, $32$, and $36$. It must be odd, because otherwise one of the numbers above or below the $20$ would be a multiple of $4$. Checking every possibility doesn't actually take very long, and you can't go very far using any other number but $11$.

That gets you to

Partial solution

Then

I made a couple of informed guesses, using the information that the $3$ and the $5$ can't be in the same group, that the $40$ must be part of a group summing to $100$, and similar things. Wrong guesses quickly lead to contradictions at this point.

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