riddle of the day

It has the feel of an ancient problem done right: a merchant, a balance scale, and the need for perfect precision. The beauty here is not in testing combinations one by one, but in discovering the system that makes them all work.

The merchant's weights

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Numerical territory

Riddle statement

A merchant wants to weigh any whole number of kilograms from 1 to 40 using only four weights.

He has a two-pan balance scale and may place weights on either side of the scale, including on the same side as the object being weighed.

What should the four weights be?

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Solution

Answer: The weights must be 1, 3, 9 and 27 kilos.

Explanation:

As the weights can be placed on both plates, each one can contribute in three ways:

  • not using it;
  • put it on the plate opposite the object, which is equivalent to adding its weight;
  • put it on the same plate as the object, which is equivalent to subtracting it.

Each weight works like a "digit" with values ​​\(-1\), \(0\) or \(1\). The natural basis for this type of representation is that of the powers of 3:

\[ 1,\ 3,\ 9,\ 27. \]

With these four weights any integer between 1 and 40 can be expressed as a combination of the form

\[ a_0\cdot 1+a_1\cdot 3+a_2\cdot 9+a_3\cdot 27, \qquad a_i\in\{-1,0,1\}. \]

That is exactly the ternary balanced.

Some examples:

  • \(2 = 3-1\),
  • \(8 = 9-1\),
  • \(20 = 27-9+3-1\),
  • \(40 = 27+9+3+1\).