riddle of the day

Ten results, no labels, and five weights to reconstruct.

Five boxes, ten weighings

Strategist
Numerical territory

Riddle statement

Five boxes have unknown weights.

Every possible pair was weighed exactly once. The note recording which two boxes produced each result was then lost, but these ten weights, in kilograms, were preserved:

20, 26, 32, 33, 39, 42, 45, 48, 54, and 61

Can you determine with certainty how much each box weighs?

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Solution

Answer: The boxes weigh 7, 13, 19, 26, and 35 kg.

Let their weights, from lightest to heaviest, be

$a\le b\le c\le d\le e.$

The sum of the ten recorded weighings is

$20+26+32+33+39+42+45+48+54+61=400.$

Each box appears in exactly four pairs. Therefore,

$4(a+b+c+d+e)=400,$

so the total weight of all five boxes is

$a+b+c+d+e=100.$

The smallest result must come from the two lightest boxes:

$a+b=20.$

The largest must come from the two heaviest:

$d+e=61.$

This isolates the middle weight:

$c=100-20-61=19.$

The second-smallest result must be \(a+c\):

$a+c=26,$

so

$a=7,\qquad b=13.$

Likewise, the second-largest result must be \(c+e\):

$c+e=54,$

and therefore

$e=35,\qquad d=26.$

Thus the five weights are

$\boxed{7,\ 13,\ 19,\ 26,\text{ and }35\text{ kg}}.$

Checking every pair reproduces exactly the ten results in the puzzle.