4
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The objective is to enter the numbers 1,2,3,4,5,6,7,8,9,10 in the blue circles. Doing this, the sum of the numbers has to be equal in the three connected rectangles (a rectangle consists of 4 connected circles, my painting skills aren't the best) !

Also, try to determine the smallest possible sum in one rectangle and try to prove it mathematically.

enter image description here

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4
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I've got:

20, which is minimal.

Like this:

    9     8     10
 4     2     3      1
    5     7     6

Minimality proof:

Let $R$ be the sum of a rectangle, and let $x$ and $y$ be the values of the two repeated circles. $1+\dots+10=55$. Then we have $3R=55+x+y$. This tells us $R\gt18$ and also $x+y\equiv 2\pmod3$, of which $x+y=5$ is the first possible and $x,y=2,3$ is a possibility. Above is an example, so $R=20$ is minimal.

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  • 1
    $\begingroup$ Great ! If you want you can prove it mathematically ? $\endgroup$ – Matti Jun 21 at 19:54

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