# Puzzles

## 24 December

There are six 3-digit numbers with the property that the sum of their digits is equal to the product of their digits. Today's number is the largest of these numbers.

## 23 December

Arrange the digits 1-9 in a 3×3 square so the 3-digits numbers formed in the rows and columns are the types of numbers given at the ends of the rows and columns.
The number in the first column is today's number.

a multiple of 4 | |||

a cube | |||

a multiple of 3 | |||

today's number | a cube | an odd number |

## 22 December

In bases 3 to 9, the number 112 is:
\(11011_3\),
\(1300_4\),
\(422_5\),
\(304_6\),
\(220_7\),
\(160_8\), and
\(134_9\).
In bases 3, 4, 6, 8 and 9, these representations contain no digit 2.

There are two 3-digit numbers that contain no 2 in their representations in all the bases between 3 and 9 (inclusive). Today's number is the smaller of these two numbers.

## 21 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10.
Today's number is the smallest number you can make with the digits in the red boxes.

+ | - | = 7 | |||

÷ | - | ÷ | |||

+ | ÷ | = 8 | |||

× | × | × | |||

+ | - | = 7 | |||

= 12 | = 5 | = 28 |

## 20 December

The integers from 2 to 14 (including 2 and 14) are written on 13 cards (one number per card). You and a friend take it in turns to take one of the numbers.

When you have both taken five numbers, you notice that the product of the numbers you have collected is equal to the product of the numbers that your friend has collected.
What is the product of the numbers on the three cards that neither of you has taken?

## 18 December

The final round of game show starts with £1,000,000. You and your opponent take it in turn to take any value between £1 and £900.
At the end of the round, whoever takes the final pound gets to take the money they have collected home, while the other player leaves with nothing.

You get to take an amount first. How much money should you take to be certain that you will not go home with nothing?

## 17 December

Eve picks a three digit number then reverses its digits to make a second number. The second number is larger than her original number.

Eve adds her two numbers together; the result is 584. What was Eve's original number?

## 16 December

Arrange the digits 1-9 in a 3×3 square so that:
the median number in the first row is 6;
the median number in the second row is 3;
the mean of the numbers in the third row is 4;
the mean of the numbers in the second column is 7;
the range of the numbers in the third column is 2,
The 3-digit number in the first column is today's number.

median 6 | |||

median 3 | |||

mean 4 | |||

today's number | mean 7 | range 2 |