The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind

Week 21 (#ulink_7a647005-8dd7-5e6f-be47-4578f5bcdf2e)

141. Joey’s and Zoë’s sums

Joey calculated the sum of the largest and smallest two-digit numbers that are multiples of three. Zoë calculated the sum of the largest and smallest two-digit numbers that are not multiples of three.

What is the difference between their answers?

[SOLUTION] (#litres_trial_promo)

142. When is the party?

Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?

[SOLUTION] (#litres_trial_promo)

143. A multiple of 11

The eight-digit number ‘1234d678’ is a multiple of 11.

Which digit is d?

[SOLUTION] (#litres_trial_promo)

144. Two squares

ABCD is a square. P and Q are squares drawn in the triangles ADC and ABC, as shown.

What is the ratio of the area of the square P to the area of the square Q?

[SOLUTION] (#litres_trial_promo)

145. Proper divisors

Excluding 1 and 24 itself, the positive whole numbers that divide into 24 are 2, 3, 4, 6, 8 and 12. These six numbers are called the proper divisors of 24.

Suppose that you wanted to list in increasing order all those positive integers greater than 1 that are equal to the product of their proper divisors. Which would be the first six numbers in your list?

[SOLUTION] (#litres_trial_promo)

146. Kangaroo game

In the expression

the same letter stands for the same non-zero digit and different letters stand for different digits.

What is the smallest possible positive integer value of the expression?

[SOLUTION] (#litres_trial_promo)

147. A game with sweets

There are 20 sweets on the table. Two players take turns to eat as many sweets as they choose, but they must eat at least one, and never more than half of what remains. The loser is the player who has no valid move.

Is it possible for one of the two players to force the other to lose? If so, how?

[SOLUTION] (#litres_trial_promo)

Week 22 (#ulink_c7102e18-9563-54bc-9859-d92be81711b7)

148. A 1000-digit number

What is the largest number of digits that can be erased from the 1000-digit number 201820182018 … 2018 so that the sum of the remaining digits is 2018?

[SOLUTION] (#litres_trial_promo)

149. Gardeners at work

It takes four gardeners four hours to dig four circular flower beds, each of diameter four metres.

How long will it take six gardeners to dig six circular flower beds each of diameter six metres?

[SOLUTION] (#litres_trial_promo)

150. Overlapping squares

The diagram shows four overlapping squares that have sides of lengths 5 cm, 7 cm, 9 cm and 11 cm.

What is the difference between the total area shaded grey and the total hatched area?

[SOLUTION] (#litres_trial_promo)

151. What can T be?

Each of the numbers from 1 to 10 is to be placed in the circles so that the sum of each line of three numbers is equal to T. Four numbers have already been entered.

Find all the possible values of T.

[SOLUTION] (#litres_trial_promo)

152. Increases of 75%

Find all the two-digit numbers and three-digit numbers that are increased by 75% when their digits are reversed.

[SOLUTION] (#litres_trial_promo)

153. Three groups

For which values of the positive integer n is it possible to divide the first 3n positive integers into three groups each of which has the same sum?