Mathematical Puzzles and Curiosities

Mathematical Puzzles and Curiosities

by Barry R. Clarke

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"Very satisfying." — Will Shortz, Crossword Editor, The New York Times. This new collection features an intriguing mix of recreational math, logic, and creativity puzzles, many of which first appeared in the author's Daily Telegraph (UK) column. Requiring only basic algebra skills, classic and new puzzles include The Monty Hall Problem, The


"Very satisfying." — Will Shortz, Crossword Editor, The New York Times. This new collection features an intriguing mix of recreational math, logic, and creativity puzzles, many of which first appeared in the author's Daily Telegraph (UK) column. Requiring only basic algebra skills, classic and new puzzles include The Monty Hall Problem, The Unexpected Hanging, The Shakespeare Puzzles, and Finger Multiplication.

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Mathematical Puzzles & Curiosities

By Barry R. Clarke

Dover Publications, Inc.

Copyright © 2013 Barry R. Clarke
All rights reserved.
ISBN: 978-0-486-31572-0


Mathematical Puzzles I

1. The Baffled Brewer

Burper the brewer had just received an order for three barrels of beer, containing 16 litres, 10 litres, and 7 litres. He had the correct total quantity in three equal-sized barrels, however, the barrels were all half full. His problem was to produce the desired quantities in each barrel. To help him he had three measuring jugs having capacity three litres, two litres and one litre. Since Burper was clearly in a fix, he telephoned his mother on her cellphone who gave the following advice. Each measuring jug must be used once only to transfer a full jug of beer from one barrel to another. Furthermore, each barrel must be involved in at least two transfer operations so that on each occasion, there is either again or loss of beer. What operations are required? [Solution 43]

2. Horse Play

At Dobbin's Dressage Academy, the 13 mounted horses were standing to attention around the hexagonal parade ground, with one in the centre. Each of the 13 riders had a number from 1–13, no two numbers appearing more than once, but only two riders, 4 and 12, were displaying them. Major Snodgrass, the instructor, had told them to arrange themselves so that each of the 12 straight lines of three numbers totalled 21. Can you fill in the missing numbers? [Solution 28]

3. Court Out

While the court was out, top barrister Betsy Gambols, Chancery Division, bet her adversary that he could not solve an enigma. Spotting that Betsy was attempting to obtain a psychological advantage, barrister Justin Case decided to face up to his opponent. The problem consists of three identical dice placed end to end in a line as shown. When the four missing numbers are correctly placed on the four blank faces, the numbers 1–6 will each appear once only on the three top and three front faces. The only restriction is that the total of the three front faces and the total of the four hidden inside faces are both even numbers. Can you avoid being caught out and place the four sets of spots on the four blank faces? [Solution 18]

4. Back to Class

Miss Flogginum, the mathematics teacher, was humiliating three of her pupils in front of the class. From left to right, Droopy, Dimwit and Dibdib stood with their back to the class and each had a digit from 0-9 pinned to his rear end, no digit being repeated. No pupil could see his own digit but was able to peep at the other two. The idea was that each had to make a statement about the two-digit number that would remain if he were not present. "It's the square of a whole number," said Droopy. "It's a prime number," reported Dimwit. Dibdib was bolder and surprised the class with a delightful observation. "The difference between the number and that formed by reversing the digits is three times the sum of the two digits." What digit did each pupil have? [Solution 40]

5. Sound Arithmetic

In Loudland, where everyone makes as much noise as possible, the inhabitants have three coins in their currency: a 3 honk, 4 honk, and 7 honk. One inhabitant recently bought a 2000 Watt stereo system for 63 honks using 15 coins, however, because of the noise going on around him, he couldn't recall how many of each coin type he had used. The three numbers of coins were each from 1 to 9 inclusive and no two numbers were the same. How many of each had he used? [Solution 44]

6. Digital Dilemma

Professor Crumble only had a good memory for digital facts. He once saw a five digit number which used some of the digits 1–9, no two digits being identical. He recalled that twice the third digit equalled the sum of the second and fourth ones, the fifth minus the second came to four, the third was a square number, the fifth was prime, and the fourth could be obtained by adding five to the first. Can you suggest where Professor Crumble saw this number? [Solution 1]

7. Core Conundrum

Old Mother Pip had a wooden box of apples partitioned into a 4×4 grid of compartments. Shown is the number of apples in each section. Rearrange the digits in the grid so that when completed, each row and column totals the same number. Only one row and one column is shown completely correct, with no other number correctly positioned. Can you find the correct arrangement? [Solution 14]

8. Word in the Stone

The archaeologist Doug Wither-Trowel was excavating a site in France one pleasant afternoon when he came upon an unusual stone with a four letter English word engraved upon it. In his notebook, he wrote that no two letters were identical and that if one replaced each letter in the word with a number giving its alphabetic position (A=1, B=2, etc), the total of the letters is 20. Not only that, but the sum of any three of the numbers is exactly divisible by the fourth number. When his French assistant read the entry in the notebook he came to the conclusion that the word was French. What was the English word on the stone? [Solution 5]

9. The Backward Robber

On Monday, a backward robber walked into a drugstore, pointed the gun at himself and handed the storekeeper half of the gold coins in his bag. The storekeeper, seeing his chance to make a handsome profit, demanded that the robber should also give him one third of the coins left in the bag. After counting out this number, the robber had a fit of belligerence and decided to give one half of the coins instead of one third. Exactly the same thing happened on Tuesday, Wednesday, Thursday and Friday, the robber walking into the store with the same three-digit square number of coins in his bag. By the end of the week, the storekeeper had gained a cubic number of coins. How many coins did the storekeeper receive? [Solution 59]

10. Maximum Security

A prisoner requests a transfer to a cell with the maximum possible volume but unfortunately it is also the cell with the smallest possible window. All windows are x+1 square and all cells have dimensions 3-x, 1-x, and 2-x, where x is a real number. What is the volume that no cell can exceed? [Solution 9]

Judgment Paradoxes

Hamlet and Hales v Petit

There is a scene in Shakespeare's Hamlet that appears to have borrowed from Edward Plowden's Law Reports first published in 1571, and concerns the issue as to whether or not one can die in one's own lifetime. In 1554, Dame Margaret Hales brought an action against one Cyriac Petit who had been awarded her husband's estate, forfeited by the Crown after the death of her husband Sir James Hales, a Judge of the Court of the Common Pleas. His suicide by drowning followed his implication in the plot to place Lady Jane Grey — the granddaughter of Henry VIII's younger sister Mary Tudor — on the English throne. Apparently, his imprisonment and subsequent pardon so disturbed him that he walked into a river and ended his life. The Coroner returned a verdict of felo de se, that is, he was declared a felon against himself, a self-murderer, his guilt resting on the proof that he had the 'will and intention of committing it'. The punishments inflicted under English law were the forfeiture of a Christian burial and a transfer of all property to the Crown. In this case, the Crown intended to award Cyriac Petit the forfeited estate. The case centered on whether or not the grounds for forfeiture occurred during Sir James' lifetime, for if they had not, then his wife, Dame Margaret Hales, could preserve her entitlement.

Her counsel, Serjeants Southcote and Puttrel, argued that the termination of one's own life could not be completed in one's lifetime, because as long as one was still alive, the moment of death that would consummate the suicide had not yet been attained. A felony could only be committed by a person alive, but one would first have to lose one's life before a self-murder could be attributed, so a suicide should not be classed as a felony.

He cannot be felo de se till the death is fully consummate, and the death precedes the felony and the forfeiture.

Serjeant Walsh, counsel for Petit the intended beneficiary, responded as follows:

The act consists of three parts: the first is the imagination, which is a reflection or meditation of the mind, whether or not it is convenient for him to destroy himself, and what way it can be done; the second is the resolution, which is a determination of the mind to destroy himself; the third is the perfection, which is the execution of what the mind had resolved to do. And of all the parts, the doing of the act is the greatest in the judgment of our law, and it is in effect the whole.

Lord Dyer ruled that:

the forfeiture shall have relation to the act done by Sir James Hales in his lifetime, which was the cause of his death, viz., the throwing himself into the water ... He therefore committed felony in his lifetime, although there was no possibility of the forfeiture being found in his lifetime, for until his death there was no cause of forfeiture.

Judge Browne delivered his own interpretation:

Sir James Hales was dead. And how came he by his death? It may be answered, by drowning. And who drowned him? Sir James Hales. And when did he drown him? In his lifetime. So that Sir James Hales being alive caused Sir James Hales to die, and the act of the living was the death of the dead man. And for this offence it is reasonable to punish the living man, who committed the offence, and not the dead man.

In Hamlet, 4.7.178–9, the Queen tells Laertes that his sister Ophelia, who has previously been showing symptoms of madness, has drowned. It appears not to have been deliberate, for she climbed a tree above a brook and the branch broke:

Queen.... Which time she chanted snatches of old tunes, As one incapable of her own distresse ...

In 5.1.1–22, two Clowns satirically discuss the drowning, echoing the discussion though not the outcome of the Hales v Petit case:

Clown. Is she to be buried in Christian burial, that wilfully seekes her owne salvation?

Other. ... the Crowner hath sate on her, and finds it Christian burial.

Clown. How can that be, unlesse she drowned her selfe in her owne defence?

Other. Why, 'tis found so.

Clown. It must be Se offendendo; it cannot bee else: for here lies the point; If I drowne myself wittingly, it argues an Act: and an Act hath three branches. It is an Act, to doe, and to perform; argall [therefore] she drown'd herself wittingly ... Here lies the water; good: here stands the man; good: If the man goe to this water and drowne himself; it is will he nill he, he goes; marke you that? But if the water come to him & drowne him; he drownes not himself. Argall hee that is not guilty of his owne death shortens not his owne life.

Other. But is this law?

Clown. Ay' marry is't, Crowner's Quest Law.

As for the Hales v Petit case upon which this is based, a proof of intention is hazardous. If the felony is determined by the fact that Hales walked into the river and his death resulted, then every person who goes for a swim in the river and subsequently drowns is a felon. Surely the point is whether or not Hales had a 'determination of the mind' to drown himself, but that one has the intention of committing suicide cannot necessarily be assumed from the fact that death has resulted. There are many cases of a self-harming act that might have led to death where the subject has survived to protest it was only a 'cry for help.' It is safe to assume then that there are also cases where a 'cry for help' has resulted in death without there being an intention. Serjeants Southcote and Puttrel might have achieved a better outcome pressing this point.

The Case of Protagoras

According to Plato, the Greek philosopher Protagoras (490–420BC) was the first philosopher to teach virtue professionally. As well as being an agnostic, his interests also extended to politics and legal matters. There is a legend that Protagoras struck a deal with one of his pupils so that the payment for his tuition was to be deferred until after he had won his first legal case. Unfortunately, on completing his training the student was dismayed to find that no clients were forthcoming. Even more dismayed was Protagoras who, being impatient, moved to sue his former pupil for the full amount outstanding with the confident declaration:

Either I win this suit, or you win it. If I win, you pay me according to the judgment of the court. If you win, you pay me according to our agreement. In either case I am bound to be paid.

However, his adversary had a ready counter:

If I win, then by the judgment of the court I need not pay you. If you win, then by our agreement I need not pay you. In either case I am bound not to have to pay you.

So whose argument was right?

Of course, Protagoras realizes that he has no legal grounds for bringing the case against his former student because at no time during the presentation of the arguments in court can the student be said to have already won his first case. So Protagoras knows that he must lose this argument about his former student reneging on their agreement and so is right in claiming that "If you win, you pay me according to our agreement", for the student would then have won his first case. His student is correct in thinking that "If I win, then by the judgment of the court I need not pay you" but only in the sense that at the time the arguments were presented he had not yet reneged on their agreement. However, after winning the case the agreement becomes active and he must now pay Protagoras who, if he brought the same case again would have a different argument and would certainly win. However, Protagoras's trick of forcing his student to win a case in court to receive payment might be subject to the penalty of having to pay costs when he loses. If this is so, then given that his former student fully intends to earn an income from his training then Protagoras's more profitable course might have been to wait.

Newcomb's Paradox

The following problem was devised by Dr William New-comb of the Lawrence Livermore Laboratory, University of California but was first presented to the philosophy community by Robert Nozick. A man is shown two closed boxes, A and B, on a table. It is known that A contains $1,000 whereas B contains either nothing or $1,000,000. No one knows which. The man is told that he can keep the contents of whichever boxes he chooses but only two possibilities are available to him. Either he can:

(1) Select both boxes.

(2) Select only B.

Prior to the test a superior intelligence from an alien civilization has predicted the man's choice. The Being's predictions are given to be "almost certainly" correct, resulting from a near perfect understanding of the man's thinking processes. If the Being has predicted that the man will choose both boxes then he has left B empty. However, if the Being predicts that the man will choose only B, then he has put $1,000,000 in it. Of course, the man might randomize his choice (e.g. by the toss of a coin) but in that case the Being will "almost certainly" have anticipated this and will have left B empty. In all cases A contains $1,000. These conditions are fully understood by everyone concerned. Everyone is also aware that everyone else understands them. How can the man maximize his winnings? Let us examine the consequences.

First, let us grant the premise that such a Being exists. It is evidently not a gainful course for the man to randomize his choice as the Being will "almost certainly" have predicted this and left box B empty. Since selecting box B is 'almost' necessarily connected with gaining $1,000,000 then the man might as well choose "B only". A counter argument has been given for taking both boxes, which develops as follows. At the point in time that the man is faced with his choice, box B either contains nothing or $1,000,000. The man's choice in the present cannot influence what the Being left in box B at some time in the past. So there is a chance that there is nothing in box B if the man chooses only that box. Surely, it would be better to take both boxes and be virtually guaranteed to receive at least $1000 and perhaps an extra $1,000,000. This is better than finishing with nothing at all.

However, no one is suggesting a reverse causality, only a forward causality where the man's choice in the present has been determined by the Being in the past and has influenced what the Being put in the boxes. In other words, whatever the man chooses has been calculated and the contents of the boxes have been decided by the Being on that basis. So if the Being is "almost certainly" correct then the man might as well choose box B. All objections to choosing "B only" must rely on doubt about the premises of the scenario. If we accept the predictive ability of the Being, then every single thought that the man entertains about the boxes has been accounted for by the Being. Since the choice "B only" necessitates a gain of $1,000,000 then this might as well be the selection. If we reject the assumption of the all-knowing Being then the problem is redundant.

In his 1969 paper, Nozick remarks that "To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly."


Excerpted from Mathematical Puzzles & Curiosities by Barry R. Clarke. Copyright © 2013 Barry R. Clarke. Excerpted by permission of Dover Publications, Inc..
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