# Sunday Times Teaser 2993 – Baker’s Weights

*by Peter Good*

#### Published February 02 2020 (a palindrome 02-02-2020!) (link)

A baker’s apprentice was given a 1kg bag of flour, scales and weights, each engraved with a different whole number of grams. He was told to separately weigh out portions of flour weighing 1g, 2g, 3g, and so on up to a certain weight, by combining weights on one side of the scales. He realised that he couldn’t do this if there were fewer weights, and the largest weight was the maximum possible for this number of weights, so he was surprised to find after one of these weighings that the whole bag had been weighed out. Upon investigation, he discovered that some of the weights weighed 1g more than their engraved weight. If I told you how many of the weights were too heavy, you would be able to work out what they all were.

What were the actual weights (in ascending order)?

Hi Erling

I noticed you are using a fairly narrow range to look for a solution

when you say you are considering all possible arrangements of weights:

in line 19: ie for t in range(40, 46):

Is there a logical reason why these limits are used and should it not be apparent with a comment in your code?

Geoff

Hi Geoff

Thanks for commenting.

‘All possible arrangements of weights’ refers to all the 64 possible combinations of normal and overweight weights. From (1g, 2g, 4g, 8g, 16g, 32g) -> (2g, 3g, 5g, 9g, 17g, 33g). All possible number of weighings are not considered. That is justified by the fact that the largest number of weigings to empty the bag of flour occurs when all weights are normal (former case) and smallest when all weights are overweight (latter case).

By the function rw(s, t), defined in line 11-12, we have rw((0, 0, 0, 0, 0, 0), 44) = 990 and rw((1, 1, 1, 1, 1, 1), 41) = 966, and in both cases, for the next value of t the real weights exceed 1000. And as the values increase monotonically, the range you refer to, in fact, could have been set even narrower.

(To be honest, I had those values in a spreadsheet before I wrote the programme, and that is where I (by convenience) get them from now. But that does not alter the justification above.)

Erling

PS: To Geoff: Forgot to mention that I accept the critic that it should have been justified by a comment in the code.