total gathered in every allotment was almost
the same. In fact, if the man with eleven trees had given one apple to
the man who had seven trees, and the man with fourteen trees had given
three each to the men with nine and thirteen trees, they would all have
had exactly the same.

Now, the puzzle is to discover how many apples each would have had (the
same in every case) if that little distribution had been carried out. It
is quite easy if you set to work in the right way.

97.--_The Perplexed Plumber._

When I paid a visit to Peckham recently I found everybody asking, "What
has happened to Sam Solders, the plumber?" He seemed to be in a bad way,
and his wife was seriously anxious about the state of his mind. As he had
fitted up a hot-water apparatus for me some years ago which did not lead
to an explosion for at least three months (and then only damaged the
complexion of one of the cook's followers), I had considerable regard for
him.

"There he is," said Mrs. Solders, when I called to inquire. "That's how
he's been for three weeks. He hardly eats anything, and takes no rest,
whilst his business is so neglected that I don't know what is going to
happen to me and the five children. All day long--and night too--there he
is, figuring and figuring, and tearing his hair like a mad thing. It's
worrying me into an early grave."

I persuaded Mrs. Solders to explain matters to me. It seems that he had
received an order from a customer to make two rectangular zinc cisterns,
one with a top and the other without a top. Each was to hold exactly
1,000 cubic feet of water when filled to the brim. The price was to be a
certain amount per cistern, including cost of labour. Now Mr. Solders is
a thrifty man, so he naturally desired to make the two cisterns of such
dimensions that the smallest possible quantity of metal should be
required. This was the little question that was so worrying him.

[Illustration]

Can my ingenious readers find the dimensions of the most economical
cistern with a top, and also the exact proportions of such a cistern
without a top, each to hold 1,000 cubic feet of water? By "economical" is
meant the method that requires the smallest possible quantity of metal.
No margin need be allowed for what ladies would call "turnings." I shall
show how I helped Mr. Solders out of his dilemma. He says: "That little
wrinkle you gave me would be useful to others in my trade."

98.--_The Nelson Column._