100 Numerical Games

Pierre Berloquin, Martin Gardner, Denis Dugas

• Publisher: Dover Publications
• Year: 2015
• ISBN:
  9780486805696
• Edition: 1
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

Follow the hour hand and minute hand of a clock for 24 hours. How many times do they form a right angle?
Timothy's house has several rooms, each of which has an even number of doors, including doors that lead outside. Is the number of outside doors even or odd?
Stimulating and delightful, this collection of puzzles features original and classic brainteasers. The author, a puzzle columnist for Le Monde, specially selected these mind-benders for the widest possible audience, ensuring that they're neither too hard for those without a math background nor too easy for the mathematically adept. All puzzles are clearly stated and accurately answered at the back of the book ― and they're great fun to consider, whether you crack them or not. Includes a Foreword by Martin Gardner.

SOLUTIONS

Game 1

Timothy entered each store with twice as much as the amount which is $1 more than he had when he left. Since he was broke when he left the fifth store, he entered it with:

(0 + 1) × 2 = $2

Likewise, he entered:

• the fourth store with (2 + 1) × 2 = $6;

• the third store with (6 + 1) × 2 — $14;

• the second store with (14 + 1) × 2 = $30;

• the first store with (30 + 1) × 2 = $62.

Timothy started with $62.

Game 2

At each vertex of the diamond, an A yields 1 ACE (that is, you can only get to 1 C from each A in a corner), or a total of 4.

On each side of the diamond, an A yields 2 ACE’s (that is, you can get to 2 C’s from each A on a side), or a total of 8.

Thus ACE can be read off in 12 ways.

Game 3

During the first 12 hours the minute hand is over the hour hand at 12:00 and won’t cross it again for at least an hour. In the last hour the minute hand does not cross the small one except at the twelfth hour (12:00).

However, the hands cross once during each hour in between; the 12 crossings in 12 hours determine 11 intervals, and during each interval the hands form a right angle twice. Total: 22 right angles.

There is no overlap between the 12-hour periods, as far as right angles go. There are another 22 right angles in the second 12 hours. Thus the hands form 44 right angles during 24 hours.

Game 4

Game 5

The first train met is pulling in as our train leaves the station. The last train met is pulling out as our train arrives at its destination.

In between, we meet a train every half-hour—that is, 9 trains.

Total trains met: 11.

Game 6

Game 7

At 5:00 the clock takes 6 seconds for 4 intervals between strikes: each interval is 1 1/2 seconds. The time needed to strike 9:00 confirms this calculation.

At any hour there are 1 less intervals than strikes. Intervals in 24 hours are:

2(0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11) = 132

Time needed for striking is:

1.5 × 132 = 198 seconds

Game 8

Game 9

Ten operations are needed. The table shows how many liters are in each container after each operation. In the first operation the first container is filled; in the second and third, the second container receives the water from the first, which is then refilled; in the fourth and fifth, the second container receives as much as possible from the first and is then emptied; and so on.

7-liter container	11-liter container
7	0
0	7
7	7
3	11
3	0
0	3
7	3
0	10
7	10
6	11

Game 10

Game 11

73 hens lay a total of 1 dozen eggs in. 1 day, and 37 hens eat a total of 1 kilogram of wheat in 1 day. To get a dozen eggs you must feed 73 hens for 1 day. This requires 73/37 kilograms of wheat—a little less than 2.

Game 12

Let us count the POP’s according to the position of the first P:

• at the center: P touches 4 O’s each to...