Math Puzzle - THE RUBY BROOCH.

THE RUBY BROOCH.

(Unclassified Problems.)
The annals of Scotland Yard contain some remarkable cases of jewel
robberies, but one of the most perplexing was the theft of Lady
Littlewood's rubies. There have, of course, been many greater robberies
in point of value, but few so artfully conceived. Lady Littlewood, of
Romley Manor, had a beautiful but rather eccentric heirloom in the form
of a ruby brooch. While staying at her town house early in the eighties
she took the jewel to a shop in Brompton for some slight repairs.
"A fine collection of rubies, madam," said the shopkeeper, to whom her
ladyship was a stranger.
"Yes," she replied; "but curiously enough I have never actually counted
them. My mother once pointed out to me that if you start from the centre
and count up one line, along the outside and down the next line, there
are always eight rubies. So I should always know if a stone were
missing."
Six months later a brother of Lady Littlewood's, who had returned from
his regiment in India, noticed that his sister was wearing the ruby
brooch one night at a county ball, and on their return home asked to
look at it more closely. He immediately detected the fact that four of
the stones were gone.
"How can that possibly be?" said Lady Littlewood. "If you count up one
line from the centre, along the edge, and down the next line, in any
direction, there are always eight stones. This was always so and is so
now. How, therefore, would it be possible to remove a stone without my
detecting it?"
"Nothing could be simpler," replied the brother. "I know the brooch
well. It originally contained forty-five stones, and there are now only
forty-one. Somebody has stolen four rubies, and then reset as small a
number of the others as possible in such a way that there shall always
be eight in any of the directions you have mentioned."
There was not the slightest doubt that the Brompton jeweller was the
thief, and the matter was placed in the hands of the police. But the man
was wanted for other robberies, and had left the neighbourhood some time
before. To this day he has never been found.
The interesting little point that at first baffled the police, and which
forms the subject of our puzzle, is this: How were the forty-five rubies
originally arranged on the brooch? The illustration shows exactly how
the forty-one were arranged after it came back from the jeweller; but
although they count eight correctly in any of the directions mentioned,
there are four stones missing.

Answer:

In this case we were shown a sketch of the brooch exactly as it appeared
after the four rubies had been stolen from it. The reader was asked to
show the positions from which the stones "may have been taken;" for it
is not possible to show precisely how the gems were originally placed,
because there are many such ways. But an important point was the
statement by Lady Littlewood's brother: "I know the brooch well. It
originally contained forty-five stones, and there are now only
forty-one. Somebody has stolen four rubies, and then reset as small a
number as possible in such a way that there shall always be eight stones
in any of the directions you have mentioned."
The diagram shows the arrangement before the robbery. It will be seen
that it was only necessary to reset one ruby--the one in the centre. Any
solution involving the resetting of more than one stone is not in
accordance with the brother's statement, and must therefore be wrong.
The original arrangement was, of course, a little unsymmetrical, and for
this reason the brooch was described as "rather eccentric."

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THE RUBY BROOCH.

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