Super-Venn (Sums in Circles)

The following diagram is composed of 10 overlapping circles, each identified by a letter. Each letter has a different whole number value from 1 through 9; since there are two circles labeled H, H will have the same value for both circles. Where some circles overlap, there is a number indicated: This number is the sum of the values of the letters in the overlapping circles. Can you figure out the correct numerical value for each letter?


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From the diagram, we have the following equations:

A + C = 7 and C + F = 10, which together imply F = A + 3.

B + C = 13 and C + F = 10, which together imply B = F + 3.

D + G = 9 and G + J = 12, which together imply J = D + 3.

G + J = 12 and H + J = 13, which together imply H = G + 1.

H + J = 13 and E + H = 9, which together imply J = E + 4.

We notice that the letters used in the first two lines above are distinct from the letters used in the last three lines, so let’s consider each group separately. We can make a chart of possible values for the letters using the above relationships and then see which combinations fit the constraints of the problem (values between 1 and 9; different letters have different values).

Starting with A + C = 7, A and C can have values from 1 to 6, with F and then B correspondingly determined. We get:


We see there are two sets of values that could work for A, B, C, and F.

Now let’s look at D, E, G, H, and J. Starting with G + D = 9, G and D can have values from 1 to 8, with J, H, and E = J – 4 correspondingly determined. We get:


We see there are three sets of values that could work for D, E, G, H, and J, but only one set is disjoint from one of the sets we found for A, B, C, and F. So our solution is:

A = 3, B = 9, C = 4, D = 2, E = 1, F = 6, G = 7, H = 8, and J = 5.

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