Point Pairs

http://www.mscroggs.co.uk/blog/59

University College London mathematician Matthew Scroggs has released this year’s puzzle Christmas card for Chalkdust magazine.

This year’s card contains 10 puzzles, each with a numerical answer. Splitting each answer into two strings of digits will identify two dots on the card’s cover to be connected by a straight line. Drawing 10 lines correctly will produce a Christmas-themed picture.

You can download the card as a PDF or complete it online.

The Indecisive Rower

A rower rows regularly on a river, from A to B and back. He’s got into the habit of rowing harder when going upstream, so that he goes twice as fast relative to the water as when rowing downstream. One day as he’s rowing upstream he passes a floating bottle. He ignores it at first but then gradually grows curious about its contents. After 20 minutes of arguing with himself he stops rowing and drifts for 15 minutes. Then he sets out after the bottle. After some time rowing downstream he changes his mind, turns around, and makes his way upstream again. But his curiosity takes hold once more, and after 10 minutes of rowing upstream he turns and goes after the bottle again. Again he grows ashamed of his childishness and turns around. But after rowing upstream for 5 minutes he can’t stand it any longer, rows downstream, and picks up the bottle 1 kilometer from the point where he’d passed it. How fast is the current?

Click for Answer

Footwork

https://commons.wikimedia.org/wiki/File:Ad%C3%A9lard_Turgeon_en_automobile.jpg

Each day after work, Smith takes the train to the suburb where he lives, and his chauffeur meets him at the station and drives him home. One day Smith finishes work early and arrives at the suburb one hour earlier than usual. He starts walking home, and the chauffeur meets him on the road and drives him the rest of the way. This gets Smith home 10 minutes earlier than usual. How long did he walk? (Disregard the time spent in stopping and picking up Smith, and assume that the chauffeur normally arrives at the station just as the train does.)

Click for Answer

Illumination

https://commons.wikimedia.org/wiki/File:Light_up_1.svg

In the Japanese logic puzzle Akari, you’re presented with a grid of black and white squares. The goal is to place “light bulbs” into white cells until the whole grid is illuminated. Each bulb sends out rays of light horizontally and vertically, illuminating its row and column unless a black cell blocks the rays.

There are two constraints: The bulbs must not shine on one another, and each numbered black cell must bear that many bulbs (orthogonally adjacent to it) in the finished diagram. An unnumbered black cell can bear any number of bulbs.

Here’s a moderately difficult puzzle. Can you solve it?

https://commons.wikimedia.org/wiki/File:Lightup.svg
Image: Wikimedia Commons

Ones and Twos

https://www.flickr.com/photos/evoo73/3222220257
Image: Flickr

From the Kanja Otogi Zoshi (“Collection of Interesting Results”) of Nakane Genjun, 1743:

Your friend has 30 go stones. He lines them up out of your sight, placing down either one or two stones with each deposit and calling “here” so you’ll know this has been done. When all 30 stones have been placed, you have heard him say “here” 18 times. How many deposits contained one stone, and how many two?

Click for Answer

Truth Pays

https://commons.wikimedia.org/wiki/File:2005-Penny-Uncirculated-Obverse-cropped.png

Here are a penny and a quarter. Make a statement. If your statement is true, then I’ll give you one of these coins (not saying which). But if your statement is false, then I won’t give you either coin.

Raymond Smullyan says, “There is a statement you can make such that I would have no choice but to give you the quarter (assuming I keep my word).” What statement will accomplish that?

Click for Answer

Winners and Losers

A puzzle by the Hungarian-Canadian mathematician George Grätzer:

I’m writing an article about a round-robin tennis tournament, in which each player plays each other player once. I decide to pick one player and ask her which players she defeated (in tennis there are no ties). Then I’ll ask each of those players which players they’ve defeated. Is it possible to pick a player so that everyone in the tournament is mentioned in my article?

Click for Answer

Dead Reckoning

buchanan chess puzzle

A.G. Buchanan posed this curious puzzle in The Problemist in July 2001. Who moved last? The question seems absurd. If each side has only a bare king, how can we know which made the last move?

The answer turns on Rule 5.2b in the Laws of Chess:

The game is drawn when a position has arisen in which neither player can checkmate the opponent’s king with any series of legal moves. The game is said to end in a ‘dead position.’ This immediately ends the game, provided that the move producing the position was legal.

In the position above, suppose it was Black who moved last. He cannot simply have moved his king to the corner from a7 or b8, because in that earlier position Rule 5.2b would already have applied: The game would have ended in a draw at that point, and Black would have had no opportunity to move his king to a8. Similarly, Black cannot have captured a knight or a bishop on a8, because neither of those pieces (alone with a king) is sufficient to give checkmate, and again the game would have ended before the diagrammed position could be reached.

Black might have captured a rook or a queen on a8. But consider that case: Suppose there was a queen on a8, and the black king was in check on a7 or b8. In that case the capture was forced — Black had no other legal move. And hence even before the capture took place it would have been correct to say that “neither player can checkmate” — the capture was ordained and no possible mate lay in the future. And so the game would have ended at that point, and again we could never have reached the diagrammed position.

Hence Black has no possible legal last move, and the answer to the puzzle is that White moved last, capturing a black piece on c6. Because this capture wasn’t forced, Rule 5.2b is not invoked.

This is a technicality, but it’s an important one. In 2015 the World Federation of Chess Composition voted that the “dead position” rule applies only to retrograde (backward-looking) problems like the one above. More details are here.