Hand Count

http://commons.wikimedia.org/wiki/File:Yankee_Stadium_Grandstand_Level_View.jpg — Image: Wikimedia Commons

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.

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If no two people are to reach the same handshake total, then the most gregarious guy in the stadium must shake 49,999 hands, the next-most-gregarious 49,998, and so on. But that means the shyest guy would shake zero hands — which is impossible, since the gregarious guy shook every hand in the place. Alternatively, if the shyest guy shakes one hand, then to avoid any duplication the gregarious guy would have to shake 50,000 hands, which is also impossible, as there are only 49,999 hands available for him to shake. Therefore there must be at least two participants, somewhere, who have shaken the same number of hands. This is an example of the “pigeonhole principle” — which also explains why 30,000 of us have the same number of hairs on our heads.

May 1, 2010April 30, 2013 | Puzzles

“The Eiffel Tower”

By C.W. Wood. White to mate in two moves.

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1. Qe2. If the e3 pawn makes a capture, then Qxe4 is mate. If the black king takes a pawn, then the queen gives mate on the fifth rank. From D. Littlewick, “Curiosities on the Chess-Board,” Strand, January 1910.

April 29, 2010January 24, 2013 | Puzzles

Hate Mail

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.

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Merrygreek read the letter verbatim, but by pausing in the wrong places he effectively repunctuated it, giving it the opposite meaning from the one Ralph intended. Here’s the letter as it was written: 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. “Then was the fault in reading,” says the scrivener, “and not in writing.”

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A Little Crusade

White to move his king only and mate in six moves.

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1. Kc5 Kd7 2. Kb6 Ke7 3. Kb7 Kd7 4. Kb8 Ke7 5. Kc8 Kf8 6. Kd7#: From the Huddersfield College Magazine, 1877. 04/10/2014 Wait, this doesn’t work — Black has 4. … Kd8. A better solution is 1. Kc5 Kd7 2. Kb6 Ke7 3. Ka7 Kd7 4. Kb7 Ke7 5. Kc8 Kf8 6. Kd7# (Thanks, Stanley.)

April 19, 2010April 10, 2014 | Puzzles

When in Rome

What number comes next in this series?

0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, 55, 1, 0, 1

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The next number is 9. The numbers correspond to the Roman numerals hidden in English number names:

April 17, 2010January 24, 2013 | Puzzles

Road Work

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?

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Solution to Road Work: I run the first half-mile in 4 minutes and the next in 5 minutes, and continue alternating like this throughout the marathon. In this way I’ll cover any measured mile in 9 minutes. But because the “fast” segments outnumber the “slow” ones, my average pace for the whole run is less than 9 minutes per mile. This works because the course does not cover an even number of miles. From Frank Morgan, The Math Chat Book, 2000.

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April 15, 2010January 23, 2013 | Puzzles

Menagerie

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?

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There are two possible answers here, thanks to the slipperiness of the word all: 1. I have a dog, a cat, and a fish. 2. I have two Denebian Slime Devils, or two Eight-Thumbed Mongolian Rat-Weasels, or two of anything-that’s-not-a-cat-dog-or-fish. I actually prefer the second answer. (Thanks, Remy.)

April 13, 2010January 23, 2013 | Puzzles

“Four Puzzles in One”

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.

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(a) a8, (b) h1, (c) e3, (d) g7.

April 10, 2010January 23, 2013 | Puzzles

Mate in Two

By G.A.A. Walker. White to play and checkmate all six black kings simultaneously with his second move.

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1. Bb3+ d5 (forced) 2. cxd6 e.p.######: From D. Littlewick, “Curiosities on the Chess-Board,” Strand, January 1910.

April 2, 2010January 23, 2013 | Puzzles

A Mysterious Windfall

A riddle by Isaac Newton:

Four people sat down at a table to play;
They play’d all that night, and some part of next day;
This one thing observ’d, that when all were seated,
Nobody play’d with them, and nobody betted;
Yet, when they got up, each was winner a guinea;
Who tells me this riddle I’m sure is no ninny.

Who are the players?

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Musicians.

March 30, 2010January 23, 2013 | Puzzles