Is there an efficient way to work out how many factors a large number has?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Copyright © Kenneth Kelsey
The following puzzle comes from The Ultimate Book of Number Puzzles
by Kenneth Kelsey and David King, published by The Cresset Press
1992. Sadly this book, which contains 350 intriguing puzzles to
solve, is now out of print; you might just be lucky enough to find
it on a second hand bookstall.
"Your late father was, as you know, a rather eccentric
inventor," said the solicitor to Peter, "and he has bequeathed to
you the patent specification relating to the combination mechanism
for safes. It comprises thirty-two tumblers numbered from one to
thirty-two. When the combination is first set these tumblers have
to be inserted into four cog wheels in such a way that the numbers
in each wheel total 132."
"There's nothing patentable in that, surely?" asked Peter.
"No, but in addition every horizontal row of eight tumblers must
also total 132..."
"Nor in that."
"... and continue to do so even when the cog wheels are
"Good heavens! Is that possible?"
"It must be since the patent has been incorporated into several
safes. Unfortunately when your father filed the patent application
he deliberately omitted several of the numbers from the
specification drawings in order to keep the relative positions of
the tumblers a trade secret. He kept the completed drawings locked
away in his private safe together with all his securities."
"Well then," said Peter "let's look in there."
"That's the problem," said the solicitor. "We can't open the
darned thing until we have the combination!"
From the incomplete specification given in the diagram shown
above, could you open the safe? In solving this puzzle bear in mind
that there are 16 positions for each cog wheel, not eight as one
might initially believe. The constant of 132 is obtained from the
three horizontal lines of eight tumblers in all 16 positions as
well as from the sum of the eight tumblers in each cog wheel.