51¶¯Âþ

I USED to run a village shop and I had a pair of scales and a set of 10
weights. Each weight was a whole number of ounces and a total weight was less
than 1024 ounces. I found that I could put one or more weights on each of the
two pans of my scales so that they balanced.

One day I lost a weight. By chance I was able to replace it with a spare;
unfortunately it was not the same weight as the one I lost. However, now I
could put one or more weights on each of the two pans of my scales so that
they balanced; in fact, I could arrange for each side to weigh 6 ounces.
Unfortunately I could not weigh 13 ounces, although I could weigh 3
ounces.

Q1. What was my set of weights after the replacement?

Q2. Suppose I was back with my original set of weights. Could I have
selected one and replaced it by some other whole-number-of-ounces weight, of
my own choosing, so that the total weight was still less than 1024 ounces and
I still could not put one or more weights on each of the two pans of my scales
so that they balanced?

Q3. As Q1 but now I can replace six weights instead of just one. The new
set of weights must not be the same as my original set.

A £10 book token will be awarded to the sender of the first correct
answer opened on Thursday 7 March 1996. the Editor’s decision is final. Please
send entries to Enigma 860, 51¶¯Âþ, King’s Reach Tower, Stamford
Street, London SE99 0BB. The winner of Enigma 853 was J. A. Grant of Guiseley
in West Yorks.

Answer to Enigma 853

Franglais

4302568