## Hard maths riddles: Card games

A small number of cards has been lost from a complete pack. If I deal among four people, three cards remain. If I deal among three people, two remain and if I deal among five people, two cards remain. How many cards are there?

This one is pretty tricky, but the answer is: There are 47 cards. To get the answer, you need to know that there are 52 cards in a pack. Then, find the multiples of four, three, and five closest to that number, and see if any of them are the same. For example, you can deal 42, 44, and 48 cards amongst four people, which would mean 9, 5, or 1 cards is missing (with three remaining). However, the only one that works for multiples of three and five people is when five cards are missing, which gives us a total of 47 cards.

## Hard maths riddles: Knight moves

I have a calculator that can display ten digits. How many different ten-digit numbers can I type using just the 0-9 keys once each, and moving from one keypress to the next using the knight’s move in chess? (In chess, the knight moves in an L shape: one square up and two across, two squares down and one across, two squares up and one across, and other like combinations.)

You can form the numbers 5034927618 and 5038167294. You can also form their reverses: 8167294305 and 4927618305. Hence four different numbers can be made. The key point is to realize that the number must start or end on the “5” key, followed/preceded by the ‘0’ key, otherwise, there is no way of using all ten keys during the route.

## Hard maths riddles: Three’s company

You know 2 + 2 comes to the same as 2 x 2. Now find a set of three different whole numbers whose sum is equal to their total when multiplied.

The three different whole numbers whose sum is equal to their total when multiplied are 1, 2, and 3.

## Hard maths riddles: Apple harvest

Mrs Jones was very proud of her apple tree. One autumn, after harvesting her apples, she called her three sons together. “Here are 150 apples,” she said. “I want you to take them to the market tomorrow and sell them for me.” She gave Paul 15 apples, Nick 50, and Ben 85. “Your job,” added Mrs. Jones, “is to sell the apples in such a way that each of you brings home the same amount of money.” How do they do it?

The first buyer purchases 12 dozen apples at \$1 per dozen. Paul sells him one dozen and has three apples left; Nick sells him four dozen and has two apples left; and Ben sells him seven dozen and has one apple left. Then a second buyer comes along and buys all their remaining apples for \$3 apiece. The three brothers head home with \$10 each.

Time for a mental workout? Try these 20 tricky rebus puzzles.

## Hard maths riddles: Seven times

What is the smallest whole number that is equal to seven times the sum of its digits?

The answer to this maths riddle is 21. You probably just guessed to answer this maths riddle, which is fine, but you can also work it out algebraically. The two-digit number ab stands for 10a + b since the first digit represents tens and the second represents units. If 10a + b = 7(a + b), then 10a + b = 7a + 7b, and so 3a = 6b, or, more simply, a = 2b. That is, the second digit must be twice the first. The smallest such number is 21.

## Hard maths riddles: Lunch money

John noticed that the amount he was paying for his lunch was a rearrangement of the digits of the amount of money he had in his pocket, and that the money he had left over was yet another rearrangement of the same three digits! How much money did John start with?

John started with \$9.54. The money can be written with just three digits, so it must be between \$1.01 and \$9.99. Trial and error shows that there is only one set of numbers that fits this question: \$9.54 = \$4.59 + \$4.95.

## Hard maths riddles: Upside down numbers

What is the smallest number that increases by 12 when it is flipped and turned upside down?