In how many ways can 100 be written as a sum of consecutive positive even numbers

**Solution**: There are **two **ways to write 100 as a sum of consecutive positive even numbers: 22+24+26+28 and 16+18+20+22+24

In how many ways can 100 be written as a sum of consecutive positive odd numbers?

**Solution: **There are also **two **ways to write 100 as a sum of consecutive positive odd numbers: 49+51 and 1+3+5+7+9+11+13+15+17+19.

What is the product of all the even numbers between -49 and 49?

**Solution**: One of the even numbers between -49 and 49 is 0, so the product of all the numbers is **0**

Alfie draws two regular polygons P and Q. Each interior angle of P is 100 degrees more than each interior angle of Q. How many sides do P and Q have in total?

**Solution: **Suppose Q had 4 or more sides. Then each interior angle of Q would be at least 90 degrees, so each interior angle of P would be at least 190 degrees, which is impossible. Therefore Q must have 3 sides (with interior angles each 60 degrees), and P must have interior angles each 160 degrees. Then each exterior angle of P is 180-160=20 degrees, and P has 360/20=18 sides. Hence, in total, P and Q have 3+18=21 sides

Two numbers have a sum of 2 and a product of -960. What is their difference?

**Solution**: The two numbers are 32 and -30, so their difference is **62**

Two numbers have a sum of 1 and a product of -5/16. What is their difference?

**Solution: **The two numbers are 5/4 and -1/4, so their difference is **3/2 **(or **5**)

How many of the first 100 square numbers are divisible by 4?

**Solution**: If you square an even number (a multiple of 2), the result is a multiple of 2×2=4. If you square an odd number, the result is still odd, so not divisible by 4. Hence, exactly half of the first 100 square numbers are divisible by 4, that is **50 **in total.

A fair six-sided die, with faces numbered 1, 2, 3, 4, 5, 6, is rolled twice. What is the probability that the product of the two numbers rolled is greater than the sum?

**Solution: **There are 6×6=36 possible pairs of numbers that could be rolled. Of these, most have a product greater than their sum. The exceptions are when one or both numbers are equal to 1 (this occurs in 11 pairs), and when both numbers are 2 (this occurs in 1 pair). So there are 36-11-1=24 pairs for which the product is greater than the sum, and the required probability is 24/36=2/3

Arthur 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 he from where he started?

**Solution**: Arthur is 1-1/2+1/4-1/8=8/8-4/8+2/8-1/8=**5/8 of a mile north **of where he 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**.

A drawer contains 4 blue socks, 6 red socks and 8 purple socks. You randomly take socks from the drawer, arranging them into a pile. What is the smallest number of socks you must remove, to be certain that the pile contains three different colours?

**Solution**: If you take 14 socks, you might have 6 red socks and 8 purple socks, without any blue socks. But if you take 15 socks, it is impossible that they contain only 1 or 2 colours between them. So the answer is **15**

What is the fraction with lowest denominator, between 2/3 and 3/4?

**Solution: **No fractions with a denominator of 1, 2, 3, 4, 5 or 6 are between 2/3 and 3/4, but 5/7 is between 2/3 and 3/4. So the answer is 5/7

At 10:15, Freya went for a short walk, then she painted until 11:00. She walked for one quarter of the time she painted. When did she start painting?

**Solution**: Splitting 45 minutes into 5 equal parts, each part is worth 9 minutes. So she started painting at **10:24**

A drawer contains 4 blue socks, 6 red socks and 8 purple socks. You randomly take socks from the drawer, arranging them into a pile. What is the smallest number of socks you must remove, to be certain that the pile contains a matching pair?

**Solution: **If you take 3 socks, you might have 1 of each colour, but if you take 4 socks, it is impossible for them all to have different colours. So the answer is **4**

The letters A, A, N, N are written on four tiles (one letter per tile), and placed in a bag. They are then drawn out of the bag at random, one at a time, and placed in a line from left to right. What is the probability that the tiles spell ANNA?

**Solution**: The probability that the first letter is A is 1/2. With an A removed, the probability the next two letters are both N’s is 2/3 x 1/2 = 1/3. With ANN removed, the last letter is always an A. So the probability of spelling ANNA is 1/2 x 1/3 = **1/6**

What is the highest common factor of 117, 171 and 711?

**Solution: **117, 171 and 711 all have a digit sum of 9, so they are all multiples of 9. Dividing each by 9 leaves 13, 19 and 79, all of which are prime so have no common factors (other than 1). So the highest common factor is 9.

If x is 2/3 of y, and z is 4/5 of x, what fraction of z is y?

**Solution**: z=(4/5)x=(4/5)(2/3)y=(8/15)y. So y=(15/8)z. That is, y is **15/8** of z.

What is the largest three-digit multiple of 6 that is still a multiple of 6 when its digits are reversed?

**Solution: **A number is divisible by 6 precisely when it is even and its digits add to a multiple of 3. So we require the first and last digits to be even. The largest number satisfying the given conditions is **894**.

How many positive whole numbers less than 100 can be written as a sum of five different positive odd numbers?

**Solution**: The smallest is 1+3+5+7+9=25, and by repeatedly adding 2 to the largest of the five odd numbers, we can obtain every odd number up to 99 (1+3+5+7+81). The sum of five odd numbers must be odd, so no even numbers can be formed. Hence the answer is the number of odds in the list 25,27,29,…,99, which is **38**

A number n exceeds one of its square roots by 20. What are the possible values of n?

**Solution: **The two possible values of n are **25** (which is 20 greater than 5), and **16** (which is 20 greater than -4)

On a farm, the ratio of the number of pigs to number of sheep is 6:7 and the ratio of the number of sheep to number of cows is 3:8. In simplest form, what is the ratio of pigs to sheep to cows on the farm?

**Solution**: The ratio of pigs to sheep is 18:21 and the ratio of sheep to cows is 21:56. So the ratio of pigs to sheep to cows is **18:21:56**

It is possible to make three different solid cuboids with 8 unit cube blocks, with dimensions of 1x1x8, 1x2x4 and 2x2x2 respectively (two cuboids are considered the same if they have the same shape). How many different solid cuboids can be made with 60 blocks?

**Solution: **The possible dimensions are 1x1x60, 1x2x30, 1x3x20, 1x4x15, 1x5x12, 1x6x10, 2x2x15, 2x3x10, 2x5x6 and 3x4x5, so the required total is 10

An ice-cream parlour sells 5 different flavours of ice-cream. Iona wants two scoops in a bowl. How many different flavour combinations can she order? (the flavours can be the same or different)?

**Solution**: Call the flavours A,B,C,D,E. There are 10 different combinations of two scoops with different flavours (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE) and 5 different combinations with identical flavours (AA, BB, CC, DD, EE), so there are **15 **combinations in total.

Which negative number is four times as far away from 41 as from 2?

**Solution**: **-11 **(it differs from 2 by 13 and from 41 by 52)

What is the smallest positive integer whose digits have a product of 60?

**Solution**: There are no such integers with one or two digits. The required number is **256**

Alice has twice as many marbles as Bob. Then Alice gives 10 of her marbles to Bob. Now Bob has three times as many marbles as Alice. How many marbles do they have in total?

**Solution: **Alice and Bob had 16 and 8 marbles respectively before the switch, and 6 and 18 marbles respectively after the switch. In total they have **24**

Two fair dice are rolled and the two numbers obtained are added. What is the probability that this sum is divisible by 5?

**Solution**: There are 6×6=36 possible outcomes, of which 7 produce a sum divisible by 5 (1 and 4, 2 and 3, 3 and 2, 4 and 1, 4 and 6, 5 and 5, 6 and 4). So the answer is **7/36**

A Triangular prism has V vertices, E edges and F faces. What is the value of V+E+F?

**Solution**: V=6, E=9 and F=5, so V+E+F=6+9+5=**20**.

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**