Question

In the figure below, each circle will be filled with a digit from $1$ to $6.$ Each digit must appear exactly once. The sum of the digits in neighboring circles is shown in the box between them. What digit must be placed in the top circle?

$2$

$3$

$4$

$5$

it is impossible to fill the circles

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Written solution:

Let the top digit be $x$ . Moving clockwise from the top, the other digits are $9-x$ , $x-2$ , $7-x$ , $x+1$ , and $6-x$ . For these six values to be exactly $1,2,3,4,5,6$ , the value $x=5$ works and gives the set $5,4,3,2,6,1$ . Thus the top digit is $5$ .

2026 AMC 8 Problem 12

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Written solution:

Problem 12 in Other Years