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Balancing Scales

Problem

In order for the scales to balance the block on the right side must be 10 kg.


It is possible to use the same weights to weigh, for example, 4 kg.


In fact using 2 kg, 3 kg and 5 kg weights it is possible to weigh all but one value from 1 kg to 10 kg. What is that value?
(Note: You don't have to use all three weights.)


Solution

The following weights can be found, with the desired weight indicated in brackets.

Left Side  Right Side
3  2 (1)
2  (2)
3  (3)
2 5  3 (4)
5  (5)
3 5  2 (6)
2 5  (7)
3 5  (8)
2 3 5  (10)

And so it is not possible to weigh 9 kg.

What if you used 1 kg, 3 kg and 6 kg to weigh all the weights from 1 kg to 10 kg?
What about using 1 kg, 2 kg and 7 kg?
If x + y + z = T, what must be special about x, y and z so that all the weights from 1 kg to T kg can be weighed?
(Hint: Try x + y = T to begin with.)

Problem ID: 91 (Dec 2002)     Difficulty: 1 Star

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