The sum of the interior angles of an n-sided polygon is twice the sum of the interior angles of a hexagon. What is the value of n?

**Solution**: The interior angles in a hexagon sum to 4×180 degrees. So the n-sided polygon has an interior angle sum of 8×180 degrees. Hence n=**10**

Amelia walks 2 miles from her house to the park at a speed of 4 miles per hour. She runs back home at a speed of 10 miles per hour. What is her average speed for the whole journey?

**Solution: **It takes Amelia half an hour to walk to the park, and one fifth of an hour to return. That is 1/2 + 1/5 = 5/10 + 2/10 = 7/10 of an hour in total, to cover 4 miles. So her average speed is 4 / 0.7 = 40/7 = **5 5/7 miles per hour**

What is the sum of the lengths of the edges of a 3x4x5 cuboid?

**Solution**: There are four edges of each of the lengths 3, 4 and 5, so the required sum is 4(3+4+5)=**48**

Two siblings have ages in the ratio 2:3, but in three years the ratio will be 7:10. What is the sum of their ages now?

**Solution**: Their current ages are 18 and 27 (in three years their ages will be 21 and 30), so the sum of their current ages is **45**

A fair 6-sided die is rolled twice. What is the probability that the second number is higher than the first?

**Solution**: There are 15 combinations that result in the second number higher then the first, out of 36 combinations in total, so the required probability is 15/36=**5/12**

What number is three quarters of four fifths of five sixths of 170?

**Solution: **By cancelling common factors, we find that 3/4 x 4/5 x 5/6 = 3/6 = 1/2 , and half of 170 is **85**

The sum of five consecutive integers is 5. What is the sum of their squares?

**Solution**: The five integers are -1,0,1,2 and 3. So the sum of their squares is 1+0+1+4+9=**15**.

Thomas begins facing northwest. He then turns clockwise through 4050 degrees. In what direction is he now facing?

**Solution: **3600 degrees is 10 full revolutions. Another full revolution completes 3960 degrees, and 4050 degrees is only 90 degrees further. So he is now facing northeast

What is the surface area of a cube with volume 64?

**Solution**: The length of one edge is 4, so the area of one face is 16, hence the surface area is 16×6=**96**

Theo has read two fifths of a novel, and has 171 pages left. How many pages are in the book?

**Solution: **171 pages represents three fifths of the book. So one fifth is 171/3=57 pages, and five fifths (that is, the whole book) is 57×5=285 pages

A collection of 1×1 unit squares is used to construct a rectangle of area 40. What is its smallest possible perimeter?

**Solution**: The perimeter will be minimised when the side lengths are as close together as possible. This happens with side lengths of 5 and 8, so the answer is 2x(5+8)=**26**

What is the largest three-digit number divisible by 4, 6 and 8?

**Solution: **A number is divisible by 4, 6 and 8 precisely when it is divisible by 24, and the largest three-digit multiple of 24 is **984**.

A pair of gloves costs £5.50 more than a hat. Together the gloves and the hat cost £18. What is the price of the hat?

**Solution**: The hat costs **£6.25 **(the gloves cost £11.75)

In a football tournament, each of 15 teams plays every other team exactly once. How many matches are there in total?

**Solution: **Each of the 15 teams plays 14 matches, suggesting 15×14=210 matches in total. But this counts every match twice (for example, team A v team B and team B v team A). So the final total is 210/2=**105**

The price of a shirt should have been reduced by 25%, but instead it was increased by 25% to £25.50. What should the price have been changed to?

**Solution**: The original price of the shirt was 4/5 x £25.50 = £20.40, so the correct new price is £20.40 x 3/4 = **£15.30**

What fraction of the factors of 324 are odd?

**Solution: **Since 324=4×81, all the factors of 324 can be split into groups of three, of the form (d, 2d, 4d), where d is an odd factor. For example, one of the groups of three is (9, 2×9, 4×9), that is, (9,18,36). In each group of three, only the first factor is odd. So **one third **of the factors of 324 are odd.

A water jug is 2/3 full. After 50ml is added, the jug is 3/4 full. What is the capacity of the jug?

**Solution**: 50ml represents (3/4)-(2/3)=(9-8)/12=1/12 of the jug, so the capacity of the jug is 12×50=**600ml**

What is the smallest multiple of 11 with a digit sum of 11?

**Solution: **No two-digit multiples of 11 work (11 to 99), and nothing in the 100s works either. But 209=11×19, and 2+0+9=11; so the answer is **209**

Aurelia walks one mile north, then half a mile south, then half of half a mile north, then half of half of half a mile south. How far north or south is she from where she started?

**Solution**: Aurelia is 1-1/2+1/4-1/8=8/8-4/8+2/8-1/8=**5/8 of a mile north **of where she started

Which integer is closest to the cube root of 15?

**Solution: **The cube root of 8 is 2 and the cube root of 27 is 3. So the cube root of 15 is between 2 and 3. To decide which is closer, we can cube 2.5. We have 25x25x25=625×25=15625, so 2.5×2.5×2.5=15.625. Hence the cube root of 15.625 is 2.5. This means that the cube root of 15 is between 2 and 2.5. Therefore the closest integer to the cube root of 15 is **2**.

The sides of a triangle have lengths 4, 7 and k, where k is a whole number. How many possible values are there for k?

**Solution**: In a triangle, we require the sum of the lengths of the two shorter sides to be greater than the length of the third side. So the possible values of k are 4, 5, 6, 7, 8, 9 and 10; that is **7 **possible values.

What is the smallest positive integer with three square factors?

**Solution: **The first few square numbers are 1, 4, 9, 16, 25, … . The smallest number with three square factors is **16**, which is divisible by 1, 4 &16

How many square factors does 300 have?

**Solution**: The square factors of 300 are 1, 4, 25 and 100 – that is **four**

How many square numbers less than 1000 are also cube numbers?

**Solution: **Squares that are also cubes must be sixth powers (some integer raised to the power of six). There are only three such numbers less than 1000: 1, 64 and 729

On 27/09/21 the sum of the digits of the day (2+7) was equal to the number of the month (9). How many more times did this happen in 2021?

**Solution**: This happened again on 19/10, 28/10 and 29/11; that is **three **more times

Does there exist a number that is larger than its cube?

**Solution: **There exist lots of numbers larger than their cube: any number between 0 and 1, and any number less than -1; so the answer is yes

The reciprocal of a number is the result of dividing 1 by the number. For example, the reciprocal of 4 is 1/4. What is the reciprocal of 1/2 + 1/3?

**Solution**: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, and the reciprocal of 5/6 is **6/5**

A positive integer (whole number) n is divisible by 14 and 15. How many other positive integers are guaranteed to be factors of n?

**Solution: **Since n is divisible by 14 and 15, it must be divisible by the lowest common multiple of 14 and 15, which is 210. It must also be divisible by all the factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. So there are 16 factors of 210, of which we have already counted the factors 14 and 15; hence there are **14** other positive integers guaranteed to be factors of n