Place the numbers 1-16 in locations A-P such that each line has the same sum.
Solve for all possible sums.
Source: Original (though likely elsewhere, too...)
Let S the common sum to the eight lines abd a,b,c,d,e,f,...the values in
locations A,B,C,D,E,F,...
We have 8 relations such as a + m + p + f = S, a + i + n + j + d = S, b
+ n + o + e = S, etc...
By addition of these relations, we can infer 2*(a + b + c + d +....) + m
+ n + o + p = 8S.
As a + b + c + d +....+ o + p = 17*16/2 = 136, we get S = 34 + (m + n +
o + p )/8. That means that the sum of the 4 terms m, n, o and
p must be a multiple of 8.
On the other hand by eliminating successively a,b,c,d...in the eight
relations, we get a relation between the remaining variables i, j, k, l,
m, n, o and p : 2*( i + j + k + l) = m + n + o + p.
As the minimum value of i + j + k + l is 1 + 2 + 3 + 4 = 10, the minimum
value of m + n + o + p is >=2*10 = 20. The fisrt number >= 20 divisible
by 8 is 24. So the minimum value of S is 37.
At last the maximum value of of m + n + o + p is 16 + 15 + 14 + 13 = 58.
The first number <=58 which is divisible by 8 is 56. So the
maximum value of S is 41.
As a conclusion the possible values of S are 37, 38, 39, 40 and
41.
There exist many solutions for each of these values.
Hereafter, one solution is
given as an example for each possible value of S.
Sum = 37 |
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Sum = 38 |
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14 |
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15 |
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14 |
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15 |
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3 |
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8 |
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12 |
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4 |
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16 |
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37 |
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16 |
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2 |
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7 |
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13 |
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38 |
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4 |
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3 |
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9 |
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8 |
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7 |
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13 |
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37 |
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11 |
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38 |
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1 |
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1 |
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37 |
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11 |
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10 |
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37 |
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38 |
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10 |
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5 |
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38 |
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37 |
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37 |
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37 |
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37 |
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38 |
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38 |
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38 |
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38 |
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Sum = 39 |
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Sum = 40 |
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9 |
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16 |
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11 |
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1 |
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8 |
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13 |
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4 |
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14 |
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8 |
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39 |
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15 |
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14 |
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40 |
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12 |
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12 |
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15 |
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39 |
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16 |
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40 |
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3 |
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39 |
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11 |
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39 |
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40 |
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40 |
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39 |
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39 |
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39 |
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39 |
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40 |
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40 |
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40 |
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40 |
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Sum = 41 |
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41 |
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41 |
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41 |
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41 |
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41 |
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41 |
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41 |
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41 |
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