Question

Jo has baked 40 cakes for her husband's 40th birthday, one for each of his years, weighing 1, 2, 3 ... 40kg. She forgets which cake is which and has to check the weight of each cake using a large pair of balancing scales. Each cake is placed individually on the left-hand pan of the scales and balanced againt one or more scale-weights which can be put on either or both scale pans as necessary. She has four scale-weights available, each a different whole number of kg. Find their values in kg and make a list showing the particular combination of the scale-weights for identifying each cake

Answer 1

Since the weight can be put on either pan, any given weight can contribute 3 values +w, 0, -w. The weights need to be chosen to be a factor of 3 of each other:
.. 1 kg, 3 kg, 9 kg, 27 kg.

The base-3 version of the numbers 1 to 40 tell you where to put the weights.

If we use the code - = weight not used, o = weight opposite the cake, w = weight with the cake, then we can make a table, as in the attachment. Left-to-right the corresponding weights are 27 kg, 9 kg, 3 kg, 1 kg.