The Humble-Nishiyama Randomness Game

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Mathematicians Steve Humble and Yutaka Nishiyama invented this game to highlight a surprising result in probability, based on a principle discovered by Walter Penney.

Two players play the game using an ordinary deck of cards. The cards will be dealt out in a row, one after another. Before the dealing begins, each player claims an ordered sequence of colors that might turn up, for example “red black red” (RBR) or “black red red” (BRR). As the cards are dealt, if three successive cards turn up in one of these sequences, the player who claimed it gets to collect those three cards as a trick, and the dealing continues. When all 52 cards have been dealt, the player who has collected the most tricks wins. (In a typical game, 7 to 9 tricks are won.)

This sounds like a perfectly even game, but in fact the second player has a strategy that will given him a significant advantage. When the first player has chosen his sequence (say, RRB), the second player changes the middle color, adds it to the start of the sequence, discards the last color, and claims the resulting sequence (in this case, BRR). This gives a decided advantage to the second player no matter which sequence his opponent has chosen. In a computer simulation of 1,000 games, Humble and Nishiyama got these results:

BBB vs RBB – RBB wins 995 times, 4 draws, BBB wins once
BBR vs RBB – RBB wins 930 times, 40 draws, BBR wins 30 times
BRB vs BBR – BBR wins 805 times, 79 draws, RBR wins 116 times
RBB vs RRB – RRB wins 890 times, 68 draws, RBB wins 42 times
BRR vs BBR – BBR wins 872 times, 65 draws, BRR wins 63 times
RBR vs RRB – RRB wins 792 times, 85 draws, RBR wins 123 times
RRB vs BRR – BRR wins 922 times, 51 draws, RRB wins 27 times
RRR vs BRR – BRR wins 988 times, 6 draws, RRR wins 6 times

Penney’s original game uses coin flips; cards are preferable because no record-keeping is required and because the finite number of cards in a deck increases the second player’s chances.

(Steve Humble and Yutaka Nishiyama, “Humble-Nishiyama Randomness Game — A New Variation on Penney’s Coin Game,” Mathematics Today, August 2010.)

| Science & Math

An Evening Stroll

http://www.math.uwaterloo.ca/tsp/pubs/index.html

Maybe this was inevitable: A team of mathematicians have worked out the most efficient pub crawl in the United Kingdom, connecting 24,727 pubs in the shortest possible closed loop, 45,495,239 meters, or about 28,269 miles. Because it’s a loop, a determined crawler can start at any point and eventually find himself back home.

Despite the pickled application, this represents a serious achievement in computational mathematics, an advance in the so-called traveling salesman problem (TSP), which asks for the shortest route that passes through each of a set of points once and once only. The pub crawl includes more than 100 times the previous record number of stops in a road-distance TSP.

“We, of course, did not have in mind to bring everything mathematics has to bear in order to improve the lot of a wandering pub aficionado,” wrote lead researcher William Cook of the University of Waterloo. “The world has limited resources and the aim of the applied mathematics fields of mathematical optimisation and operations research is to create tools to help us to use these resources as efficiently as possible.”

(Thanks, Danesh.)

| Science & Math

First Things First

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In Languages and Their Speakers (1979), linguist Timothy Shopen shows how greetings and leave-takings can reflect a society’s cultural values. First he gives a typical American conversation in which one friend encounters another who is 15 minutes late for work:

Hello Ed!

— Hi! How are you?

Sorry, I’m in a hurry.

— Yeah, me too.

See you on Saturday.

The whole interaction lasts five seconds; it includes a greeting and a leave-taking, but there is no actual conversation between. Here’s the same interaction in the Maninka culture of West Africa:

Ah Sedou, you and the morning.

— Excellent. You and the morning.

Did you sleep in peace?

— Only peace.

Are the people of the household well?

— There is no trouble.

Are you well?

— Peace, praise Allah. Did you sleep well?

Praise Allah. You Kanté.

— Excellent. You Diarra.

Excellent.

— And the family?

I thank Allah. Is there peace?

— We are here.

How is your mother?

— No trouble.

And your cousin Fanta?

— Only peace. And your father?

Praise Allah. He greets you.

— Tell him I have heard it.

And your younger brother Amadou?

— He is well. And your uncle Sidi?

No trouble, Praise Allah …

Where are you going?

— I’m going to the market. And you?

My boss is waiting for me.

— O.K. then, I’ll see you later.

Yes, I’ll see you later. Greet the people of the household.

— They will hear it. Greet your father.

He will hear it.

— May your day pass well.

Amen. May the market go well.

— Amen. May we meet soon.

May that “soon” arrive in good stead.

“Time elapsed: 46 seconds,” Shopen writes. “It is more important to show respect for a friend or a kinsman than to be on time for work, and thus we have the example of Mamadou Diarra above, already fifteen minutes late for work and not hesitating to be even later in order to greet a friend in the proper manner. First things first, and there is no question for the Maninka people about what is most important.”

| Language · Society

The Last Blessing

https://commons.wikimedia.org/wiki/File:Mark_Twain_1909.jpg

After his daughter Jean’s death in 1909, Mark Twain began to write:

Would I bring her back to life if I could do it? I would not. If a word would do it, I would beg for strength to withhold the word. And I would have the strength; I am sure of it. In her loss I am almost bankrupt, and my life is a bitterness, but I am content: for she has been enriched with the most precious of all gifts — that gift which makes all other gifts mean and poor — death. I have never wanted any released friend of mine restored to life since I reached manhood. I felt in this way when Susy passed away; and later my wife, and later Mr. Rogers. When Clara met me at the station in New York and told me Mr. Rogers had died suddenly that morning, my thought was, Oh, favorite of fortune — fortunate all his long and lovely life — fortunate to his latest moment! The reporters said there were tears of sorrow in my eyes. True — but they were for ME, not for him. He had suffered no loss. All the fortunes he had ever made before were poverty compared with this one.

“I am setting it down,” he told his friend Albert Bigelow Paine, “everything. It is a relief to me to write it. It furnishes me an excuse for thinking.”

He wrote for three days, handed the manuscript to Paine, and told him to make it the final chapter of his autobiography. Four months later he was dead.

| Death · Literature

Floor, Please?

https://commons.wikimedia.org/wiki/File:Tri%C3%A1ngulo_de_Pascal_sin_r%C3%B3tulo.svg
Image: Wikimedia Commons

Reader Alex Freuman passed this along — a simple method of establishing any row in Pascal’s triangle, attributed to Edric Cane. To establish, for example, the seventh row (after the initial solitary 1), create a row of fractions in which the numerators are 7, 6, 5, 4, 3, 2, 1 and the denominators are 1, 2, 3, 4, 5, 6, 7:

\displaystyle \frac{7}{1} \times \frac{6}{2} \times \frac{5}{3} \times \frac{4}{4} \times \frac{3}{5} \times \frac{2}{6} \times \frac{1}{7}

Now multiply these in sequence, cumulatively, to get the numbers for the seventh row of the triangle:

\displaystyle 1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1

These are the coefficients for

\displaystyle \left ( a+b \right )^{7}=a^{7} + 7a^{6}b + 21a^{5}b^{2} + 35a^{4}b^{3} + 35a^{3}b^{4} + 21a^{2}b^{5} + 7ab^{6} + b^{7}.

Cane writes, “It couldn’t be easier to remember or to implement.” Another example — row 10:

\displaystyle \frac{10}{1} \times \frac{9}{2} \times \frac{8}{3} \times \frac{7}{4} \times \frac{6}{5} \times \frac{5}{6} \times \frac{4}{7} \times \frac{3}{8} \times \frac{2}{9} \times \frac{1}{10}

\displaystyle 1 \quad 10 \quad 45 \quad 120 \quad 210 \quad 252 \quad 210 \quad 120 \quad 45 \quad 10 \quad 1

| Science & Math

In a Word

https://commons.wikimedia.org/wiki/File:Union-rat.jpg
Image: Wikimedia Commons

obtest
v. to call heaven to witness; to protest against

proditor
n. a traitor; a betrayer

Is a union breaking the law if it posts a giant inflatable rat outside an employer’s facility? No, it’s not, according to a 2011 decision by the National Labor Relations Board. The Sheet Metal Workers’ Union had sought to dissuade a hospital from using non-union workers by stationing a 16-foot rat near the building’s entrance. The NLRB held that the “the use of the stationary Giant Rat (i) constituted peaceful and constitutionally protectable expression, (ii) did not involve confrontational conduct that would qualify as unlawful picketing, and (iii) did not qualify as nonpicketing conduct that was otherwise unlawfully coercive.”

The “rat collosi” are multiplying (gallery). Let’s hope they don’t stage an uprising themselves someday.

| Language · Society

False Start

https://commons.wikimedia.org/wiki/File:Jean_Louis_Th%C3%A9odore_G%C3%A9ricault_001.jpg

The horse raced past the barn fell.

That’s a grammatically correct sentence. What does it mean? Most readers have to puzzle over it a bit before seeing the interpretation

The horse [that was] raced past the barn fell.

This is a “garden-path sentence” — the reader naturally assumes one interpretation and is confused to find that another had been intended. Further examples:

The old man the boat.

The government plans to raise taxes were defeated.

The cotton clothing is made of grows in Alabama.

I convinced her children are noisy.

Time flies like an arrow; fruit flies like a banana.

In writing, lexicographer Henry Fowler calls it “an obvious folly — so obvious that no one commits it wittingly except when surprise is designed to amuse. But writers are apt to forget that, if the false scent is there, it is no excuse to say they did not intend to lay it; it is their business to see that it is not there, and this requires more care than might be supposed.”

| Language