Two by Two

Here’s a curious way to multiply two numbers. Suppose we want to multiply 97 by 23. Write each at the head of a column. Now halve the first number successively, discarding remainders, until you reach 1, and double the second number correspondingly in its own column:

two by two - first image

Cross out each row that has an even number in the left column, and add the numbers that remain in the second column:

two by two - second image

That gives the right answer (97 × 23 = 2231). Why does it work?

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Essentially the technique converts the first factor into binary, multiplies each of its constituents by the second factor, and sums the results. Imagine that each line is associated with a power of 2: the first line with 2⁰, the second with 2¹,and so on. The business in the first column, halving the first factor successively and crossing out those lines with even numbers, effectively reduces the first factor to its binary constituents — here, the lines that remain are those associated with 2⁰, 2⁵, and 2⁶, and, sure enough, 2⁰ + 2⁵ + 2⁶ = 97. Now we need to multiply each of those constituents by the second factor, 23. In other words, we want to find: (23 × 2⁰) + (23 × 2⁵) + (23 × 2⁶) That’s what’s accomplished in the second column. Doubling the second factor successively, as we’ve done there, is equivalent to multiplying it by 2⁰ in the first line, by 2¹ in the second, and so on. That is, 23 = 23 × 2⁰, 46 = 23 × 2¹, etc. And the lines that haven’t been crossed out are precisely the ones we want: 23 (= 23 × 2⁰) 736 (= 23 × 2⁵) 1472 (= 23 × 2⁶) So if we add those values, we’ll get the product of the original two numbers, which is what we sought: 23 + 736 + 1472 = 2231. Here’s essentially what we’ve done, from the top: 97 × 23 = (2⁰ + 2⁵ + 2⁶) × 23 = (23 × 2⁰) + (23 × 2⁵) + (23 × 2⁶) = 23 + 736 + 1472 = 2231 It works with any pair of numbers.

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July 18, 2009January 21, 2013 | Puzzles · Science & Math

Editorializing

A marble-cutter, inscribing the words,–‘Lord, she was thine’ upon a tombstone, found that he had not figured his spaces correctly and he reached the end of the stone one letter short. The epitaph therefore read:

‘Lord, she was thin.’

— Frederic William Unger, Epitaphs, 1904

July 14, 2009July 14, 2009 | Death · Humor

Right and Wrong

Brother Jacques Percher, “a very excellent man of the old time,” had a painting made for his chapel showing that good is the very opposite of evil. At one side was a picture of an angel, with the words “Read the right side and you will be saved.” Under that was this inscription:

Delicias fuge, ne frangaris crimine, verum
Coelica tu quaeras, ne male dispereas,
Respicias tua, non cujusvis quaerito gesta
Carpere, sed laudes, nec preme veridicos.
Judicio fore te praesentem conspice toto
Tempore, nec Christum, te rogo, despicias:
Salvificum pete, nec secteris daemonia; Christum
Dilige, nequaquam tu mala concupito.

Shun pleasures of the flesh, lest you be broken by crime; seek the things of heaven, lest your end be an evil one; consider your own deeds, and do not seek to slander someone else’s, but praise them, and do not suppress those who speak the truth; always realize that you must stand before a judgment; I beg you, do not despise Christ, seek him who gives salvation, and do not follow the devil; love Christ, and do not lust at all after evil.

At the other side was a picture of the devil with the words “Read the wrong side and you will be damned.” Here the first inscription was reversed word for word, producing an entirely different meaning:

Concupito mala tu, nequaquam dilige Christum,
Daemonia secteris, nec pete salvificum;
Despicias, rogo te, Christum, nec tempore toto
Conspice praesentem te fore judicio:
Veridicos preme, nec laudes, sed carpere gesta
Quaerito cujusvis, non tua respicias,
Dispereas male, nec quaeras tu coelica; verum
Crimine frangaris, ne fuge delicias.

Lust after evil, and do not at all love Christ; you follow the devil, do not seek him who gives salvation; despise Christ, I beg you, and realize that never will you stand before a judgment; suppress those who speak the truth, and do not praise the deeds of anyone, but seek to slander them; do not consider your own; let your end be an evil one, do not seek the things of heaven; let yourself be broken by crime, do not shun pleasures of the flesh.

“It must have taken the brother a long time to compose this,” writes George Wakeman, “but he probably did it with a holy purpose, and as a recreation from more onerous duties.”

Applied Chemistry

Oxyphenbutazone is a metabolite of phenylbutazone.

It’s also the biggest goldmine in Scrabble — play it across three Triple Word Score squares, joining eight tiles that have already been played and extending seven existing words, and you’ll earn 1,780 points with a single play.

A Curious Conversation

You’re standing with your friends Val and Colin when a stranger approaches and shows you 16 cards:

A♥ Q♥ 4♥
J♠ 8♠ 7♠ 4♠ 3♠ 2♠
K♣ Q♣ 6♣ 5♣ 4♣
A♦ 5♦

He shuffles the cards, selects one, and tells Val the card’s value and Colin the card’s color. Then he asks, “Do you know which card I have?”

Val says, “I don’t know what the card is.”

Colin says, “I knew that you didn’t know.”

Val says, “I know the card now.”

Colin says, “I know it too.”

What is the card?

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It’s the ace of diamonds. If Val couldn’t name the card knowing only its value, then it must be an ace, a queen, a 5, or a 4, as these are the only values that appear more than once. If Colin knew that Val couldn’t name it, it must be red, as all the red cards have values (A, Q, 5, 4) that appear more than once. When Val understands that the card must be red, she can exclude from her list of possibilities the queen, the 5, and the 4, as each of those cards appears in a black suit. That leaves only the ace of diamonds. Adapted from Julian Havil’s Impossible?, 2008. UPDATE: Wait, this is cooked. Havil gives the problem but not the solution, and my solution above isn’t valid. Thanks, Ben and Andrew, for pointing it out. If anyone can find a valid solution, let me know and I’ll post it here. I don’t think one is possible. FWIW, this is based on a more famous “impossible puzzle,” which is soluble but spectacularly hard. UPDATE 2: In Canada or England, does the word “colour” mean “suit”? If so, the problem is soluble. It seems to have appeared originally on page 14 of a paper [PDF] by University of Victoria philosopher Audrey Yap. The first two steps above are right — Val can deduce that the card is red and must be A, Q, 5, or 4. The fact that she’s able to say “I know the card now” at this point means it can’t be an ace (because that value appears in both red suits). And the fact that Colin can then say “I know it too” means that it must be the five of diamonds. So, from the top: Val says, “I don’t know what the card is.” (She knows it’s a 5, but there are two 5s.) Colin says, “I knew that you didn’t know.” (He knows it’s a diamond, and thus an A or 5; but as each of these values appears more than once in the deck, he knows that Val won’t have enough information yet to identify the suit.) Val says, “I know the card now.” (Colin’s remark has just told her that the card is red, because all the red cards in this deck have values that appear in multiple suits. And the only red 5 is 5♦.) Colin says, “I know it too.” (He knows that Val was looking for an A or a 5, and that learning the card’s color was helpful. Thus she can’t have been looking for an A, as both aces are red. That leaves 5♦.) So, if “colour” means “suit,” the answer is 5♦. If “colour” means “color,” then I still think it’s insoluble. Thanks to the many smart people who wrote in about this, especially Helge.

Click for Answer

It’s the ace of diamonds.

If Val couldn’t name the card knowing only its value, then it must be an ace, a queen, a 5, or a 4, as these are the only values that appear more than once.

If Colin knew that Val couldn’t name it, it must be red, as all the red cards have values (A, Q, 5, 4) that appear more than once.

When Val understands that the card must be red, she can exclude from her list of possibilities the queen, the 5, and the 4, as each of those cards appears in a black suit.

That leaves only the ace of diamonds.

Adapted from Julian Havil’s Impossible?, 2008.

UPDATE: Wait, this is cooked. Havil gives the problem but not the solution, and my solution above isn’t valid. Thanks, Ben and Andrew, for pointing it out. If anyone can find a valid solution, let me know and I’ll post it here. I don’t think one is possible.

FWIW, this is based on a more famous “impossible puzzle,” which is soluble but spectacularly hard.

UPDATE 2: In Canada or England, does the word “colour” mean “suit”? If so, the problem is soluble. It seems to have appeared originally on page 14 of a paper [PDF] by University of Victoria philosopher Audrey Yap. The first two steps above are right — Val can deduce that the card is red and must be A, Q, 5, or 4. The fact that she’s able to say “I know the card now” at this point means it can’t be an ace (because that value appears in both red suits). And the fact that Colin can then say “I know it too” means that it must be the five of diamonds. So, from the top:

Val says, “I don’t know what the card is.” (She knows it’s a 5, but there are two 5s.)

Colin says, “I knew that you didn’t know.” (He knows it’s a diamond, and thus an A or 5; but as each of these values appears more than once in the deck, he knows that Val won’t have enough information yet to identify the suit.)

Val says, “I know the card now.” (Colin’s remark has just told her that the card is red, because all the red cards in this deck have values that appear in multiple suits. And the only red 5 is 5♦.)

Colin says, “I know it too.” (He knows that Val was looking for an A or a 5, and that learning the card’s color was helpful. Thus she can’t have been looking for an A, as both aces are red. That leaves 5♦.)

So, if “colour” means “suit,” the answer is 5♦. If “colour” means “color,” then I still think it’s insoluble. Thanks to the many smart people who wrote in about this, especially Helge.

June 3, 2009September 24, 2017 | Puzzles

Don’t Even Try

White or Black to play and mate or self-mate in one move. That is, you must find a total of four moves from this position: a White move that mates Black instantly, a White move that forces Black to mate White instantly, and equivalent moves for Black.

“Memo: The above puzzle depends on a literal interpretation of the rule which provides that a Pawn on reaching the eighth square may become any piece irrespective of colour.”

WARNING: “This monstrosity is the production of an erratic solver who has been sorely tried, puzzled and perplexed all the year round by the many posers and problems which have appeared from time to time in the numerous Chess columns. His aesthetic patience, resignation, fortitude, culture and hope all at once breaking down, he set to work and with wrathful spirit, regardless of all problem construction, devised it more for the sake of retaliation and revenge than to give pleasure. To prove his spiteful character; when composing it, he was overheard repeating, ‘Since I cannot prove a lover to entertain these fair spoken days, I am determined to prove a villain.’ Consequently, gentle reader, we warn you not to attempt it, except indeed that you are the happy possessor of that knowledge wherein you are able to puzzle others. It may look beastly simple, but to any young solver who may be foolhardy enough to venture it we offer a few words of advice–carefully study the above memo and note that–but ‘hold enough,’ no more can we divulge, fearful of bringing the fiery wrath of the exasperated composer upon our devoted heads.”

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White: To mate, 1. Rb3#. To self-mate, 1. bxc8, promoting to a black king, pinning the pieces on b8 and c7 and leaving Black with only 1. … Kxc3#. Black: To mate, 1. … Kxc3#. To self-mate, 1. … Nxd5+, forcing 2. Nxd5#. “Notwithstanding the memo, many young solvers (and old ones too) have had a hard wrestle with this production, but all to no purpose, it was the first of its kind published. The consequence was that many communications were received by the composer doubting his sanity, denouncing the problem as a fraud and threatening that if they caught him on the other side of the channel they would give him a taste of their wrath. The receiver of these communications thought that the channel was a mighty fine affair!” From Thomas B. Rowland, Chess Fruits, 1884.

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May 28, 2009January 21, 2013 | Puzzles

Epimenides Soused

Somebody had told me of a dealer in gin who, having had his attention roused to the enormous waste of liquor caused by the unsteady hands of drunkards, invented a counter which, through a simple set of contrivances, gathered into a common reservoir all the spillings that previously had run to waste. … It struck me, therefore, on reviewing this case, that the more the people drank, the more they would titubate, by which word it was that I expressed the reeling and stumbling of intoxication. … [T]he more they titubated, the more they would spill; and the more they spilt, the more, it is clear, they did not drink. … Yet, again, if they drank nothing worth speaking of, how could they titubate? Clearly they could not; and, not titubating, they could have had no reason for spilling, in which case they must have drunk the whole–that is, they must have drunk to the whole excess imputed, which doing, they were dead drunk, and must have titubated to extremity, which doing, they must have spilt nearly the whole. … ‘And so round again,’ as my lord the bishop pleasantly expresses it, in secula seculorum.

— Thomas de Quincey, Essays on Philosophical Writers, 1856

May 26, 2009May 26, 2009 | Science & Math

Sound Reasoning

Wintering in the Canadian Arctic in 1822, Capt. W.E. Parry made a series of experiments to see whether cold affects the velocity of sound. He marked a line of 5,645 feet on the sea ice, put a six-pounder gun at one end, and stood with a second observer at the other end. The gun fired 15 blank charges, and the observers timed the interval between each flash and its report. Generally they got good results, giving a mean velocity of 1,023 feet per second. But, writes Mr. Fisher:

The Experiments on the 9th February, 1822, were attended with a singular circumstance, which was–the officers’ word of command ‘fire,’ was several times distinctly heard both by Captain Parry and myself, about one beat of the chronometer [half a second] after the report of the gun; from which it would appear, that the velocity of sound depended in some measure upon its intensity.

“The word ‘fire’ was never heard during any of the other experiments; upon this occasion the night was calm and clear, the thermometer 25° below zero, the barometer 28.84 inches, which was lower than it had ever been observed before at Winter Island.” The phenomenon, whatever it was, has never been observed elsewhere, but Parry noted another acoustic oddity on his next voyage.

May 15, 2009September 11, 2010 | Oddities · Science & Math