The above line of figures does not appear very interesting at first sight, but if one asks some charming member of the fair sex to turn it upside down and hold it to a mirror to read it, a hidden meaning becomes apparent.
— Strand, December 1908
Here’s proof that one leg of a triangle always equals the sum of the other two.
ABC is our triangle. Extend it make a parallelogram, as shown, and divide the parallelogram into a grid. Obviously,
AB + BC = (AG + HJ + KL + MN) + (GH + JK + LM + NC).
Now let the grid grow increasingly fine: Instead of dividing the parallelogram into a 4×4 grid, make it 5×5, then 6×6, and so on. With each iteration, the stairstep figure described above will approximate AC more closely, and yet its total length will always equal AB + BC. Thus, at the limit, AB + BC = AC. Where is the error?
(From Henry Dudeney’s Canterbury Puzzles, via W.W. Rouse Ball’s Mathematical Recreations and Essays, 1892.)
By Eric Angelini, Europe Echecs, 1990.
White adds one square at the edge of the board and then mates in two.
Another puzzle from Sam Loyd:
“How fast those children grow!” remarked Grandpa. “Tommy is now twice as old as Maggie was when Tommy was six years older than Maggie is now, and when Maggie is six years older than Tommy is now their combined ages will equal their mother’s age then, although she is now but forty-six.” How old is Maggie?
A tangram paradox by Sam Loyd. Each of these gentlemen is assembled from the same seven pieces. Yet one has a foot and the other doesn’t. How is this possible?
A classic railroad shunting puzzle. The segment at the top can accommodate either freight car, but not the locomotive. The freight cars can be joined if desired, and the locomotive can push or pull either or both cars from either direction. The task is to use the locomotive to swap the positions of the two cars.
From Henry Dudeney:
A better class of puzzle is the well-known one of the Railway. If New York and San Francisco are just seven days’ journey apart, and if trains start from both ends every day at noon, how many trains coming in an opposite direction will a train leaving New York meet before it arrives at its destination at San Francisco?
George Berkeley proposed that material things exist only because they are perceived to exist. In 1993, L.C. Rodó offered a suitably Berkeleyan puzzle in El Acertijo. Any piece, including a king, that is not regarded (attacked or guarded) by another piece disappears from the board. In the diagram above, every man is either attacked or guarded. But a single move (it may be by either White or Black) will set in motion a chain reaction in which all the pieces vanish from the board. What is the move?
From this position, how can White force Black to checkmate him in two moves?
Very little is known about Theophilus Thompson, a chess problemist who was born a slave in Maryland in 1855. That’s a pity, because his work is often beautiful. Here’s a sample:
White to mate in two.
Another retrograde chess puzzle by Raymond Smullyan. The pawn on g3 is of unknown color. During the game no piece has moved from a light square to a dark one or vice versa. Is the unknown pawn black or white?
A conundrum by John F. Collins, from the August 1968 issue of Word Ways:
Just then, someone came up from behind and put his hands over the Hatter’s eyes.
‘Guess who!’ said the newcomer in a thin, flat voice.
The Hatter froze for a moment and declared, rather coldly, ‘I have no use for practical jokers.’
‘Ha! Neither have I,’ retorted the stranger, still keeping his hands over the Hatter’s eyes.
At that, the Hatter seemed to accept the challenge of the game and started asking a series of questions in a manner that mingled hope with care.
Question: ‘Ahem. Would you, by chance, be in a black suit this evening?’
Answer: ‘I would, but not by chance, by design.’
Q. ‘I presume you’re a member of all the posh clubs?’
A. ‘Afraid not. Never even been invited.’
Q. ‘Surely you’re better than average?’
A. ‘Yes, indeed!’
Q. ‘Not spotted, I hope?’
A. ‘Knock wood.’
A. ‘No, happy.’
Who is behind the Mad Hatter?
From a 19th-century British broadside:
The love and tenderness I have hitherto expressed to
you is false, and I now feel that my indifference towards
you increases every day, and the more I see you the more
you appear ridiculous in my eyes, and contemptible–
I feel inclined and in every respect disposed and determined
to hate you. Believe me I never had any inclination
to offer you my hand. Our last conversation I assure you
left a tedious and wretched insipidity which has not
possessed me with an exalted opinion of your character,
your inconstant temper would make me miserable,
and if ever we are united, I shall experience nothing but
the hatred of my parents, added to everlasting dis-
pleasure in living with you. I have a true heart to bestow,
but however I do not wish you for a moment to think
it is in your service, as I could not give it to one more
inconstant and capricious than yourself, and one less
capable to do honour to my choice, and my family.
You, Madam, I beg and desire will be persuaded that I
think seriously, and you will do me a great favour to
avoid me. I shall excuse you taking the trouble to
give me an answer to this, as your letters are full of
nonsense and impertinence, and have not a shadow of
wit and good sense. Adieu, and believe me truly, I am
so averse to you, that it is impossible I should ever be,
Madam, your Affectionate Servant and Lover, R.G.
“By reading every other line of the above letter the true meaning will be found out.”
Raymond Smullyan devised this puzzle in 1957 while a student at the University of Chicago. The white king has just been knocked off the board during a legal game. Where was it standing?
From Henry Dudeney:
A banker in a country town was walking down the street when he saw a five-dollar bill on the curb. He picked it up, noted the number, and went to his home for luncheon. His wife said that the butcher had sent in his bill for five dollars, and, as the only money he had was the bill he had found, he gave it to her, and she paid the butcher. The butcher paid it to a farmer in buying a calf, the farmer paid it to a merchant who in turn paid it to a laundry woman, and she, remembering that she owed the bank five dollars, went there and paid the debt.
The banker recognized the bill as the one he had found, and by that time it had paid twenty-five dollars worth of debts. On careful examination he discovered that the bill was counterfeit. What was lost in the whole transaction, and by whom?
Suppose we fill Yankee Stadium with 50,000 people and ask them to spend the day shaking hands with one another.
Prove that, at the end of the day, at least two participants will have shaken hands with the same number of people.
In Ralph Roister Doister (1553), Ralph asks a scrivener to compose a love letter to Dame Christian Custance. But when Matthew Merrygreek reads it for her, Dame Custance is shocked to hear an insulting diatribe. This is certainly not what Ralph intended, but the scrivener confirms that he copied the letter accurately, and Merrygreek read it verbatim and in full. What’s going on here?
Sweet mistress, whereas I love you nothing at all–
Regarding your substance and richness chief of all–
For your personage, beauty, demeanour and wit,
I commend me unto you never a whit.–
Sorry to hear report of your good welfare,
For (as I hear say) such your conditions are,
That ye be worthy favour of no living man,
To be abhorred of every honest man,
To be taken for a woman inclined to vice,
Nothing at all to virtue giving her due price.–
Wherefore, concerning marriage, ye are thought
Such a fine paragon, as ne’er honest man bought.–
And now by these presents I do you advertise
That I am minded to marry you in no wise.–
For your goods and substance, I can be content
To take you as ye are. If ye mind to be my wife,
Ye shall be assured, for the time of my life,
I will keep you right well from good raiment and fare–
Ye shall not be kept but in sorrow and care–
Ye shall in no wise live at your own liberty.
Do and say what ye lust, ye shall never please me;
But when ye are merry, I will be all sad;
When ye are sorry, I will be very glad;
When ye seek your heart’s ease, I will be unkind;
At no time in me shall ye much gentleness find,
But all things contrary to your will and mind,
Shall be done–otherwise I will not be behind
To speak. And as for all them that would do you wrong,
I will so help and maintain, ye shall not live long–
Nor any foolish dolt shall cumber you but I.
I, whoe’er say nay, will stick by you till I die.
Thus, good mistress Custance, the Lord you save and keep;
From me, Roister Doister, whether I wake or sleep–
Who favoureth you no less, ye may be bold,
Than this letter purporteth, which ye have unfold.
What number comes next in this series?
0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, 55, 1, 0, 1
I set out to run a 26.5-mile marathon hoping to average less than 9 minutes per mile. As I run, my friends measure my time along various mile-long segments of the course. On each mile that they measure — indeed, on each mile that it’s possible to measure, starting anywhere along the course — my time is exactly 9 minutes. Yet my average for the whole marathon is less than 9 minutes per mile.
How is this possible?
How many pets do I have if all of them are dogs except two, all are cats except two, and all are fish except two?
By Sam Loyd. Place the black king (a) where it can be checkmated on the move, (b) where it’s in stalemate, (c) where it’s in checkmate, and (d) where the three white pieces can’t be arranged to checkmate it.