A Dice Puzzle

Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.

The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they’re different. Their chances of winning are equal.

The first die has 5 red faces and 1 blue face. What are the colors on the second die?

Click for Answer

Weighty Matters

From the Second All Soviet Union Mathematical Competition, Leningrad 1968:

On a teacher’s desk sits a balance scale, on which are a set of weights. On each weight is the name of at least one student. As each student enters the classroom, she moves all the weights that bear her name to the other side of the scale.

Before any students enter, the scale is tipped to the right. Prove that there’s some set of students that you can let into the room that will will tip the scale to the left.

Click for Answer

Stretch Goals

stretch goals puzzle

Two circles intersect. A line AC is drawn through one of the intersection points, B. AC can pivot around point B — what position will maximize its length?

Click for Answer

Dirty Work

https://commons.wikimedia.org/wiki/File:Elsheimer,_Adam_-_Frankfurter_Kreuzaltar,_Predellentafel_%22Die_Ausgrabung_der_Kreuze%22_-_1603-1605.jpg

A puzzle by Pierre Berloquin:

Timothy, Urban, and Vincent are digging identical holes in a field.

  • When Timothy and Urban work together, they dig 1 hole in 4 days.
  • When Timothy and Vincent work together, they dig 1 hole in 3 days.
  • When Urban and Vincent work together, they dig 1 hole in 2 days.

Working alone, how long does it take Timothy to dig one hole?

Click for Answer

A Poetic Puzzle

On Oct. 25, 1875, Lewis Carroll sent this verse to Mrs. J. Chataway, mother of one of his child-friends, Gertrude. “They embody, as you will see, some of my recollections of pleasant days at Sandown”:

Girt with a boyish garb for boyish task,
Eager she wields her spade — yet loves as well
Rest on a friendly knee, the tale to ask
That he delights to tell.

Rude spirits of the seething outer strife,
Unmeet to read her pure and simple spright,
Deem, if you list, such hours a waste of life,
Empty of all delight!

Chat on, sweet maid, and rescue from annoy
Hearts that by wiser talk are unbeguiled!
Ah, happy he who owns that tenderest joy,
The heart-love of a child!

Away, fond thoughts, and vex my soul no more!
Work claims my wakeful nights, my busy days:
Albeit bright memories of that sunlit shore
Yet haunt my dreaming gaze!

He asked her leave to have it published. The child in the verse is not named — why should he feel obliged to ask permission?

Click for Answer

Seeking Stability

You’re standing in a room with an uneven floor. Before you is a square table with four legs. The table wobbles, but by turning it gradually you manage to find a position in which all four feet are supported, eliminating the wobble (though now the tabletop isn’t level).

You wonder: Is this always possible? Assuming that the four legs are of equal length and that the surface of the floor varies smoothly, is it always possible to position a four-legged table so that all four legs are supported?

Click for Answer

The Conway Immobilizer

Three positions, “left,” “middle,” and “right,” are marked on a table. Three cards, an ace, a king, and a queen, lie face up in some or all three of the positions. If more than one card occupies a given position then only the top card is visible, and a hidden card is completely hidden; that is, if only two cards are visible then you don’t know which of them conceals the missing card.

Your goal is to have the cards stacked in the left position with the ace on top, the king in the middle, and the queen on the bottom. To do this you can move one card at a time from the top of one stack to the top of another stack (which may be empty).

The problem is that you have no short-term memory, so you must design an algorithm that tells you what to do based only on what is currently visible. You can’t recall what you’ve done in the past, and you can’t count moves. An observer will tell you when you’ve succeeded. Can you devise a policy that will meet the goal in a bounded number of steps, regardless of the initial position?

“It’s tricky to design an algorithm that makes progress, avoids cycling, and doesn’t do something stupid when it’s about to win,” wrote Dartmouth mathematician Peter Winkler in sharing this puzzle in his book Mathematical Puzzles: A Connoisseur’s Collection (2003). It’s called “The Conway Immobilizer” because it originated with legendary Princeton mathematician John H. Conway and because it’s said to have immobilized one solver in his chair for six hours.

Click for Answer