
MEASURING,
WEIGHING, AND PACKING PUZZLES.
"Measure still for measure." Measure for Measure, v. 1.
Apparently the first printed puzzle involving the measuring of a given
quantity of liquid by pouring from one vessel to others of known capacity was
that propounded by Niccola Fontana, better known as "Tartaglia" (the stammerer),
15001559. It consists in dividing 24 oz. of valuable balsam into three equal
parts, the only measures available being vessels holding 5, 11, and 13 ounces
respectively. There are many different solutions to this puzzle in six
manipulations, or pourings from one vessel to another. Bachet de Méziriac
reprinted this and other of Tartaglia's puzzles in his Problèmes plaisans et
délectables (1612). It is the general opinion that puzzles of this class can
only be solved by trial, but I think formulæ can be constructed for the solution
generally of certain related cases. It is a practically unexplored field for
investigation.
The classic weighing problem is, of course, that proposed by Bachet. It
entails the determination of the least number of weights that would serve to
weigh any integral number of pounds from 1 lb. to 40 lbs. inclusive, when we are
allowed to put a weight in either of the two pans. The answer is 1, 3, 9, and 27
lbs. Tartaglia had previously propounded the same puzzle with the condition that
the weights may only be placed in one pan. The answer in that case is 1, 2, 4,
8, 16, 32 lbs. Major MacMahon has solved the problem quite generally. A full
account will be found in Ball's Mathematical Recreations (5th
edition).
Packing puzzles, in which we are required to pack a maximum number of
articles of given dimensions into a box of known dimensions, are, I believe, of
quite recent introduction. At least I cannot recall any example in the books of
the old writers. One would rather expect to find in the toy shops the idea
presented as a mechanical puzzle, but I do not think I have ever seen such a
thing. The nearest approach to it would appear to be the puzzles of the jigsaw
character, where there is only one depth of the pieces to be adjusted.
362.—THE
WASSAIL BOWL.
One Christmas Eve three Weary Willies came into possession of what was to
them a veritable wassail bowl, in the form of a small barrel, containing exactly
six quarts of fine ale. One of the men possessed a fivepint jug and another a
threepint jug, and the problem for them was to divide the liquor equally
amongst them without waste. Of course, they are not to use any other vessels or
measures. If you can show how it was to be done at all, then try to find the way
that requires the fewest possible manipulations, every separate pouring from one
vessel to another, or down a man's throat, counting as a manipulation.
Solution 363.—THE
DOCTOR'S QUERY.
"A curious little point occurred to me in my dispensary this morning," said a
doctor. "I had a bottle containing ten ounces of spirits of wine, and another
bottle containing ten ounces of water. I poured a quarter of an ounce of spirits
into the water and shook them up together. The mixture was then clearly forty to
one. Then I poured back a quarterounce of the mixture, so that the two bottles
should again each contain the same quantity of fluid. What proportion of spirits
to water did the spirits of wine bottle then contain?"
Solution 364.—THE
BARREL PUZZLE.
The men in the illustration are disputing over the liquid contents of a
barrel. What the particular liquid is it is impossible to say, for we are unable
to look into the barrel; so we will call it water. One man says that the barrel
is more than half full, while the other insists that it is not half full. What
is their easiest way of settling the point? It is not necessary to use stick,
string, or implement of any kind for measuring. I give this merely as one of the simplest
possible examples of the value of ordinary sagacity in the solving of puzzles.
What are apparently very difficult problems may frequently be solved in a
similarly easy manner if we only use a little common sense.
Solution 365.—NEW
MEASURING PUZZLE.
Here is a new poser in measuring liquids that will be found interesting. A
man has two tenquart vessels full of wine, and a fivequart and a fourquart
measure. He wants to put exactly three quarts into each of the two measures. How
is he to do it? And how many manipulations (pourings from one vessel to another)
do you require? Of course, waste of wine, tilting, and other tricks are not
allowed.
Solution 366.—THE
HONEST DAIRYMAN.
An honest dairyman in preparing his milk for public consumption employed a
can marked B, containing milk, and a can marked A, containing water. From can A
he poured enough to double the contents of can B. Then he poured from can B into
can A enough to double its contents. Then he finally poured from can A into can
B until their contents were exactly equal. After these operations he would send
the can A to London, and the puzzle is to discover what are the relative
proportions of milk and water that he provides for the Londoners'
breakfasttables. Do they get equal proportions of milk and water—or two parts
of milk and one of water—or what? It is an interesting question, though,
curiously enough, we are not told how much milk or water he puts into the cans
at the start of his operations.
Solution 367.—WINE
AND WATER.
Mr. Goodfellow has adopted a capital idea of late. When he gives a little
dinner party and the time arrives to smoke, after the departure of the ladies,
he sometimes finds that the conversation is apt to become too political, too
personal, too slow, or too scandalous. Then he always manages to introduce to
the company some new poser that he has secreted up his sleeve for the occasion.
This invariably results in no end of interesting discussion and debate, and puts
everybody in a good humour.
Here is a little puzzle that he propounded the other night, and it is
extraordinary how the company differed in their answers. He filled a wineglass
half full of wine, and another glass twice the size onethird full of wine. Then
he filled up each glass with water and emptied the contents of both into a
tumbler. "Now," he said, "what part of the mixture is wine and what part water?"
Can you give the correct answer?
Solution 368.—THE
KEG OF WINE.
Here is a curious little problem. A man had a tengallon keg full of wine and
a jug. One day he drew off a jugful of wine and filled up the keg with water.
Later on, when the wine and water had got thoroughly mixed, he drew off another jugful and
again filled up the keg with water. It was then found that the keg contained
equal proportions of wine and water. Can you find from these facts the capacity
of the jug?
Solution 369.—MIXING
THE TEA.
"Mrs. Spooner called this morning," said the honest grocer to his assistant.
"She wants twenty pounds of tea at 2s. 4½d. per lb. Of course we
have a good 2s. 6d. tea, a slightly inferior at 2s.
3d., and a cheap Indian at 1s. 9d., but she is very
particular always about her prices."
"What do you propose to do?" asked the innocent assistant.
"Do?" exclaimed the grocer. "Why, just mix up the three teas in different
proportions so that the twenty pounds will work out fairly at the lady's price.
Only don't put in more of the best tea than you can help, as we make less profit
on that, and of course you will use only our complete pound packets. Don't do
any weighing."
How was the poor fellow to mix the three teas? Could you have shown him how
to do it?
Solution 370.—A
PACKING PUZZLE.
As we all know by experience, considerable ingenuity is often required in
packing articles into a box if space is not to be unduly wasted. A man once told
me that he had a large number of iron balls, all exactly two inches in diameter,
and he wished to pack as many of these as possible into a rectangular box
24^{9}/_{10} inches long, 22^{4}/_{5} inches
wide, and 14 inches deep. Now, what is the greatest number of the balls that he
could pack into that box?
Solution 371.—GOLD
PACKING IN RUSSIA.
The editor of the Times newspaper was invited by a high Russian
official to inspect the gold stored in reserve at St. Petersburg, in order that
he might satisfy himself that it was not another "Humbert safe." He replied that
it would be of no use whatever, for although the gold might appear to be there,
he would be quite unable from a mere inspection to declare that what he saw was
really gold. A correspondent of the Daily Mail thereupon took up the
challenge, but, although he was greatly impressed by what he saw, he was
compelled to confess his incompetence (without emptying and counting the
contents of every box and sack, and assaying every piece of gold) to give any
assurance on the subject. In presenting the following little puzzle, I wish it
to be also understood that I do not guarantee the real existence of the gold,
and the point is not at all material to our purpose. Moreover, if the reader
says that gold is not usually "put up" in slabs of the dimensions that I give, I
can only claim problematic licence.
Russian officials were engaged in packing 800 gold slabs, each measuring 12½
inches long, 11 inches wide, and 1 inch deep. What are the interior dimensions
of a box of equal length and width, and necessary depth, that will exactly
contain them without any space being left over? Not more than twelve slabs may
be laid on edge, according to the rules of the government. It is an interesting
little problem in packing, and not at all difficult.
Solution 372.—THE
BARRELS OF HONEY.
Once upon a time there was an aged merchant of Bagdad who was much respected
by all who knew him. He had three sons, and it was a rule of his life to treat
them all exactly alike. Whenever one received a present, the other two were each
given one of equal value. One day this worthy man fell sick and died,
bequeathing all his possessions to his three sons in equal shares.
The only difficulty that arose was over the stock of honey. There were
exactly twentyone barrels. The old man had left instructions that not only
should every son receive an equal quantity of honey, but should receive exactly
the same number of barrels, and that no honey should be transferred from barrel
to barrel on account of the waste involved. Now, as seven of these barrels were
full of honey, seven were halffull, and seven were empty, this was found to be
quite a puzzle, especially as each brother objected to taking more than four
barrels of, the same description—full, halffull, or empty. Can you show how
they succeeded in making a correct division of the property?
Solution
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