## 20 to 3

Prove that, if we choose nine vertices of a regular icosagon, some three of these will form an isosceles triangle.

## The Froggy Problem

Speaking of Lewis Carroll — and further to Wednesday’s logic exercise — here’s the king of all Carroll’s logic problems. What’s the strongest conclusion that can be drawn from these premises?

- When the day is fine, I tell Froggy “You’re quite the dandy, old chap!”
- Whenever I let Froggy forget that 10 pounds he owes me, and he begins to strut about like a peacock, his mother declares “He shall
*not*go out a-wooing!” - Now that Froggy’s hair is out of curl, he has put away his gorgeous waistcoat.
- Whenever I go out on the roof to enjoy a quiet cigar, I’m sure to discover that my purse is empty.
- When my tailor calls with his little bill, and I remind Froggy of that 10 pounds he owes me, he does
*not*grin like a hyena. - When it is very hot, the thermometer is high.
- When the day is fine, and I’m not in the humor for a cigar, and Froggy is grinning like a hyena, I never venture to hint that he’s quite the dandy.
- When my tailor calls with his little bill and finds me with an empty pocket, I remind Froggy of that 10 pounds he owes me.
- My railway shares are going up like anything!
- When my purse is empty, and when, noticing that Froggy has got his gorgeous waistcoat on, I venture to remind him of that 10 pounds he owes me, things are apt to get rather warm.
- Now that it looks like rain, and Froggy is grinning like a hyena, I can do without my cigar.
- When the thermometer is high, you need not trouble yourself to take an umbrella.
- When Froggy has his gorgeous waistcoat on, but is
*not*strutting about like a peacock, I betake myself to a quiet cigar. - When I tell Froggy that he’s quite a dandy, he grins like a hyena.
- When my purse is tolerably full, and Froggy’s hair is one mass of curls, and when he is not strutting about like a peacock, I go out on the roof.
- When my railways shares are going up, and when it’s chilly and looks like rain, I have a quiet cigar.
- When Froggy’s mother lets him go a-wooing, he seems nearly mad with joy, and puts on a waistcoat that is gorgeous beyond words.
- When it is going to rain, and I am having a quiet cigar, and Froggy is not intending to go a-wooing, you had better take an umbrella.
- When my railway shares are going up, and Froggy seems nearly mad with joy,
*that*is the time my tailor always chooses for calling with his little bill. - When the day is cool and the thermometer low, and I say nothing to Froggy about his being quite the dandy, and there’s not the ghost of a grin on his face, I haven’t the heart for my cigar!

Unfortunately, Carroll died before he was able to publish the solution — but he warned that it contains “a beautiful ‘trap.’”

## Made to Order

1. Find an expression for the number 1 that uses each of the digits 0-9 once.

2. Do the same for the number 100.

3. Write 31 using only the digit 3 five times.

4. Express 11 with three 2s.

5. Express 10 with two 2s.

6. Express 1 with three 8s.

7. Express 5 with two 2s.

## Animal Behavior

A logic exercise by Lewis Carroll. What conclusion can be drawn from these premises?

- Animals are always mortally offended if I fail to notice them.
- The only animals that belong to me are in that field.
- No animal can guess a conundrum unless it has been properly trained in a Board-School.
- None of the animals in that field are badgers.
- When an animal is mortally offended, it rushes about wildly and howls.
- I never notice any animal unless it belongs to me.
- No animal that has been properly trained in a Board-School ever rushes about wildly and howls.

## Dog’s Mead

This puzzle was devised by W.T. Williams. The goal is to discover the age of Father Dunk’s mother-in-law (2 down), but the clues contain so much cross-reference and the grid so many interlocking solutions that practically the whole puzzle must be completed to find it.

The year is 1939. There are 20 shillings in a pound; 4840 square yards in an acre; a quarter of an acre in a rood; and 1760 yards in a mile.

ACROSS

1. Area in square yards of Dog’s Mead

5. Age of Martha, Father Dunk’s aunt

6. Difference in yards between length and breadth of Dog’s Mead

7. Number of roods in Dog’s Mead times 8 down

8. The year the Dunks acquired Dog’s Mead

10. Father Dunk’s age

11. Year of Mary’s birth

14. Perimeter in yards of Dog’s Mead

15. Cube of Father Dunk’s walking speed in miles per hour

16. 15 across minus 9 down

DOWN

1. Value in shillings per rood of Dog’s Mead

2. Square of the age of father Dunk’s mother-in-law

3. Age of Mary, father Dunk’s other daughter

4. Value in pounds of Dog’s Mead

6. Age of Ted, father Dunk’s son, who will be twice the age of his sister Mary in 1945

7. Square of the breadth of Dog’s Mead

8. Time in minutes it takes Father Dunk to walk 4/3 times round Dog’s Mead

9. The number which, when multiplied by 10 across, gives 10 down

10. See 9 down

12. Addition of the digits of 10 down plus 1

13. Number of years Dog’s Mead has been in the Dunk family

Hint: Start with 15 across, and keep your wits about you — most of the clues require some sort of insight or intelligent narrowing of the possible solutions; there’s very little mechanical plugging of numbers. With enough careful, dogged reasoning, it’s possible to complete the entire grid, but it’s stupendously hard.

## Short-Handed

Suppose we have a clock whose hour and minute hands are identical. How times times per day will we find it impossible to tell the time, provided we always know whether it’s a.m. or p.m.?

## Ice Water

A boat floats in a swimming pool. In the boat is a block of ice. If the block is dumped into the water and melts, does the water level rise or fall?

## The Broken Deck

When eight certain cards are removed from a standard poker deck, it becomes impossible to deal a straight flush. What are the cards? (Assume the deck contains no jokers.)

## Bootstrap Percolation

On a 12 × 12 grid, some squares are infected and some are healthy. On each turn, a healthy square becomes infected if it has two or more infected orthogonal neighbors. (In the example above, the black squares are infected, the white squares are healthy, and the gray squares will be infected next turn.) What’s the smallest number of initially infected squares that can spread an infection over the whole board?

## Counterfeit Redux

A more challenging version of the Counterfeit Coin puzzle from 2011:

You have 12 coins, one of which has been replaced with a counterfeit. The false coin differs in weight from the true ones, but you don’t know whether it’s heavier or lighter. How can find it using three weighings in a pan balance?

## “A Financial Puzzle”

This, now, is straightforward and business-like: A. applied to B. for a loan of $100. B. replied, ‘My dear A., nothing would please me more than to oblige you, and I’ll do it. I haven’t $100 by me, but make a note and I’ll indorse it, and you can get the money from the bank.’ A. proceeded to write the note. ‘Stay,’ said B., ‘make it $200. I want $100 myself.’ A. did so, B. indorsed the paper, the bank discounted it, and the money was divided. When the note became due, B. was in California, and A. had to meet the payment. What he is unable to cipher out is whether he borrowed $100 of B., or B. borrowed $100 of him.

– Henry C. Percy, *Our Cashier’s Scrap-Book*, 1879

## Caller ID

In his 1936 collection *Brush Up Your Wits*, British puzzle maven Hubert Phillips relates that his brother-in-law felt himself cursed with an unintelligent maid. “I have just overheard her taking a ‘phone call,” he told Phillips, “and this is what I heard:

“‘Is Mr. Smith at home?’

“‘I will ask him, sir. What name shall I give him?’

“‘Quoit.’

“‘What’s that, sir?’

“‘Quoit.’

“‘Would you mind spelling it?’

“‘Q for quagga, U for umbrella, O for omnibus, I for idiot –’

“‘I for what, sir?’

“‘I for idiot, T for telephone. Q, U, O, I, T, Quoit.’

“‘Thank you, sir.’”

Why did he accuse her of unintelligence?

## Brood War

A newlywed couple are planning their family. They’d like to have four children, a mix of girls and boys. Which is more likely: (1) two girls and two boys or (2) three children of one sex and one of the other? (Assume that each birth has an equal chance of being a boy or a girl.)

## Flip Sum

A problem from the 1999 St. Petersburg City Mathematical Olympiad:

Fifty cards are arranged on a table so that only the uppermost side of each card is visible. Each card bears two numbers, one on each side. The numbers range from 1 to 100, and each number appears exactly once. Vasya must choose any number of cards and flip them over, and then add up the 50 numbers now on top. What’s the highest sum he can be sure to reach?

## Black and White

By Peder Andreas Larsen. White to mate in two moves.

(I’ve added a guide to chess notation.)