Math Notes

Prove that log₁₀ 5 is irrational.

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Suppose it’s rational. Then log₁₀ 5 = a/b, where a and b are positive integers. Then 10^a/b = 5 and 10^a = 5^b. But these two values can never be equal — any positive integral power of 10 will end in 0, and any such power of 5 will end in 5. So log₁₀ 5 cannot be rational.

April 10, 2014April 13, 2014 | Puzzles

Black and White

carpenter chess problem

By George Edward Carpenter. White to mate in two moves.

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1. Qh2! threatens 2. Nf5#. If 1. … Kf4 then 2. Ne4#, and if 1. … Kd4 then the queen mates from h8. From The Chess Players’ Chronicle, 1874.

April 4, 2014April 13, 2014 | Puzzles

The Suicidal King

Imagine a 1000 x 1000 chessboard on which a white king and 499 black rooks are placed at random such that no rook threatens the king. And suppose the king goes bonkers and wants to kill himself. Can he reach a threatened square in a finite number of moves if Black is trying actively to avoid this?

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Yes. Imagine that the king is on the lower left square and that on each move he takes a step along the main diagonal toward the upper right. After his first move and Black’s response, there must be no rooks on the lowest three ranks or the leftmost three files, or the king could immediately reach a threatened square. After his 998th move, the king arrives at a similar situation in the upper right: At this point there must be no rooks on the uppermost three ranks or the rightmost three files, for the same reason. This means that when the king begins his journey, all the rooks must be northeast of him, and when he finishes, all must be southwest. During the trip, then, each of Black’s 499 rooks must change both its rank and its file, in order to keep out of his way. This requires 499 x 2 = 998 moves. So, at the latest, Black will run out of time just a half-move too soon: White will squeeze Black’s maneuvering room down to zero on his own 998th move, just before Black evacuates the last rook. From Miodrag Petkovic, Mathematics and Chess, 1997.

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March 26, 2014April 6, 2014 | Puzzles

Black and White

agnel chess problem

By Hyacinth R. Agnel. White to mate in two moves.

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1. Qc7! cxd2 2. Qc3# From American Chess-Nuts, 1868.

March 21, 2014March 30, 2014 | Puzzles

Black and White

shinkman chess problem

By William Anthony Shinkman. White to mate in two moves.

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If White could pass his turn, Black would have to promote the pawn or play 1. … Kd1, inviting mate in every variation: 2. Kd1 Qxd2# 2. d1=Q(or R) Bh4# 2. d1=B (or N) Bd2# To reach these lines, White needs to play a waiting move now. But this is tricky to find: If he marks time by moving the bishop, then Black can mark time by advancing the g-pawn. And if White moves the king to b3, b2, b1, or a1, then Black can promote the pawn with check. The only safe move is 1. Ka3!, and now Black must submit to one of the lines above. From the St. John Globe, 1905.

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March 14, 2014March 23, 2014 | Puzzles

Trivium

The names of two U.S. state capitals end with the same eight letters. What are they?

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Jefferson City, Missouri, and Carson City, Nevada.

March 9, 2014March 16, 2014 | Puzzles · Trivia

Black and White

gold chess problem

By Sámuel Gold. White to mate in two moves.

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With 1. Qf3 White threatens 2. Qh5#. If Black captures the queen then 2. d4 is mate. His only other defenses require moving the bishop (Bxb3/Be6/Bf7), which permits an immediate 2. Qxe4#. From The Chess Players’ Chronicle, 1872.

March 7, 2014March 16, 2014 | Puzzles

“Anglo-Foreign Words”

Walter Penney of Greenbelt, Md., offered this poser in the August 1969 issue of Word Ways: The Journal of Recreational Linguistics. Below are five groups of English words. Each group appears also in a foreign language. What are the languages?

aloud, angel, hark, inner, lover, room, taken, wig
alas, atlas, into, manner, pore, tie, vain, valve
ail, ballot, enter, four, lent, lit, mire, taller
banjo, chosen, hippo, pure, same, share, tempo, tendon
ago, cur, dare, fur, limes, mane, probe, undo

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Dutch: aloud — very old, angel — sting, hark — rake, inner — collector, lover — foliage, room — cream, taken — jobs, wig — wedge Finnish: alas — down, atlas — satin, into — enthusiasm, manner — mainland, pore — bubble, tie — road, vain — only, valve — awake French: ail — garlic, ballot — bumpkin, enter — to graft, four — oven, lent — slow, lit — bed, mire — gun sight, teller — to send out roots Japanese: banjo — sovereign, chosen — challenge, hippo — style of calligraphy, pure — pulley, same — shark, share — witticism, tempo — supplement, tendon — bowl of rice and fish Latin: ago — I act, cur — why, dare — to give, fur — thief, limes — boundary, mane — early in the morning, probe — thoroughly, undo — wave

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March 6, 2014March 16, 2014 | Language · Puzzles

Projective Chess

In projective geometry, every family of parallel straight lines intersects at an infinitely distant point. Chess problem composers in the former Yugoslavia have adapted this idea for the chessboard, adding four special squares “at infinity.”

projective chess 1

Now a queen on a bare board, for example, can zoom off to the west (or east) and reach a square “at infinity” from which she attacks every rank on the board simultaneously from both directions. She might also zoom to the north (or south) to reach a different square at infinity; from this one she attacks every file simultaneously, again from both directions. Finally she can zoom to the northwest or southeast and attack all the diagonals parallel to a8-h1, or zoom to the northeast or southwest and attack all the diagonals parallel to a1-h8. These four “infinity squares,” plus the regular board, make up the field of play.

N. Petrovic created the problem below, published in Matematika Na Shahmatnoi Doske. White is to play and mate in at least two moves. Can you find the solution?

projective chess 2

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After 1. Kg1 Black has no safe move: If he moves the king to e4 or f3, then the queen zooms westward to infinity and mates, attacking simultaneously from left and right on every rank. If he moves the king onto the g-file then the queen zooms southwestward to infinity, attacking simultaneously along all the southwest-northeast diagonals from both directions. If he moves the d-pawn then again the queen can mate from the southwest/northeast. If he moves the b6 knight (anywhere), then the queen zooms northwest and attacks along the northwest-southeast diagonals. If he moves the c7 knight then the queen zooms north and attacks along the files. Given all this firepower, White’s main task is really to find a square where his king is safely out of the way. But that’s tricky. After 1. Kh1?, for example, Black has 1. … Nbd5! and there’s no mate.

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March 5, 2014March 16, 2014 | Puzzles

Reunion

A Russian problem from the 1999 Mathematical Olympiad:

A father wants to take his two sons to visit their grandmother, who lives 33 kilometers away. His motorcycle will cover 25 kilometers per hour if he rides alone, but the speed drops to 20 kph if he carries one passenger, and he cannot carry two. Each brother walks at 5 kph. Can the three of them reach grandmother’s house in 3 hours?

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Yes. The father sets out with his first son, leaving the second boy to walk alone. Father and son drive 24 kilometers, which takes 6/5 hours. Then the father puts down the first son and rides back to meet the second son 9 kilometers from home, which takes 3/5 hours. Finally he and the second son ride 6/5 hours more and reach grandmother’s house just as the first son is arriving there on foot. Each son spends 6/5 hours riding 24 kilometers and 9/5 hours walking 9 kilometers, so the three of them reach grandmother’s in exactly 3 hours.

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March 3, 2014March 9, 2014 | Puzzles