# Puzzles

## 9 December

Arrange the digits 1-9 in a 3×3 square so that:
all the digits in the first row are odd;
all the digits in the second row are even;
all the digits in the third row are multiples of 3;
all the digits in the second column are (strictly) greater than 6;
all the digits in the third column are non-prime.
The number in the first column is today's number.

all odd | |||

all even | |||

all multiples of 3 | |||

today's number | all >6 | all non-prime |

## 8 December

Carol uses the digits from 0 to 9 (inclusive) exactly once each to write five 2-digit even numbers, then finds their sum. What is the largest number she could have obtained?

## 7 December

The sum of the coefficients in the expansion of \((x+1)^5\) is 32. Today's number is the sum of the coefficients in the expansion of \((2x+1)^5\).

## 6 December

Noel's grandchildren were in born in November in consecutive years. Each year for Christmas, Noel gives each of his grandchildren their age in pounds.

Last year, Noel gave his grandchildren a total of £208. How much will he give them in total this year?

## 5 December

28 points are spaced equally around the circumference of a circle. There are 3276 ways to pick three of these points.
The three picked points can be connected to form a triangle. Today's number is the number of these triangles that are isosceles.

## 4 December

There are 5 ways to tile a 3×2 rectangle with 2×2 squares and 2×1 dominos.

Today's number is the number of ways to tile a 9×2 rectangle with 2×2 squares and 2×1 dominos.

## 3 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 largest number you can make with the digits in the red boxes.

+ | + | = 21 | |||

+ | × | × | |||

+ | + | = 10 | |||

+ | ÷ | × | |||

+ | + | = 14 | |||

= 21 | = 10 | = 14 |

## 2 December

You have 15 sticks of length 1cm, 2cm, ..., 15cm (one of each length). How many triangles can you make by picking three sticks and joining their ends?

Note: Three sticks (eg 1, 2 and 3) lying on top of each other does not count as a triangle.

Note: Rotations and reflections are counted as the same triangle.