The mean (average) height of a class of 9 pupils is 150cm. A 10th pupil then joins the class, increasing the average to 151.5cm. How tall is the 10th pupil?
Solution: The sum of the heights of the first 9 pupils is 150×9=1350cm, and the sum of the heights of all 10 pupils is 1515cm. So the 10th pupil is 1515-1350=165cm
There is a magic plant called the illusium. When an illusium is uprooted, 10 illusium plants grow in its place. In a large field there is a single illusium. No more illusiums will be planted in the field. Is it possible that, eventually, the field will contain exactly 999 illusiums?
Solution: It is not possible – every time an illusium is uprooted, the total number if illusiums in the field increases by exactly 9. So the number of illusiums is always 1 more than a multiple of 9; but 999 is itself a multiple of 9.
A palindrome is a number that reads the same forwards and backwards, like 77 and 30903. What is the largest three-digit palindrome divisible by 6?
Solution: A number is divisible by 6 if it is even and its digits sum to a multiple of 3. The largest three-digit palindrome with this property is 888
In how many ways can five people sit in a line on a bench, if two of the people, Alice and Bob, must sit together?
Solution: For the moment, treat Alice and Bob as a single two-headed person. Then there are 4 ‘people’ to arrange in a line, which can be done in 4x3x2x1=24 ways. For each of these 24 arrangements, there are two ways to arrange Alice and Bob – Alice can go to the right or to the left of Bob. So the total number of arrangements of all five people is 24×2=48
Divide 111,111,111 by 1,001,001 without a calculator.
Solution: We have 111,111,111=100,100,100+10,010,010+1,001,001=1,001,001(100+10+1)=1,001,001×111. So the answer is 111
Insert two pairs of brackets into the expression 1-2-3-4 to make the result as large as possible.
Solution: There are five ways to bracket 1-2-3-4. The way that produces the largest result is 1-((2-3)-4)=6
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.
How many factors of 333 contain at least one digit 3?
Solution: The factors of 333 containing a digit 3 are 3, 37 and 333 itself; that is three factors
Which positive integer(s) below 100 has/have the most factors containing at least one digit 3?
Solution: There are three positive integers below 100 hat have the most factors containing a digit 3: 39, 78 and 93 each have exactly three factors containing a digit 3.
In a non-leap year, what is the date of the middle day of the year?
Solution: There are 365 days in a non-leap year, so the middle day is day 183. There are 31+28+31+30+31+30=181 days in the first six months; so the middle day is 2nd July
Convert 10,000 minutes into days, hours and minutes.
Solution: There are 60 minutes in 1 hour, and 24×60=1440 minutes in a day. So there are 6×1440=8640 minutes in 6 days, leaving 10,000-8640=1360 minutes. This is 80 minutes less than 1 day. So 10,000 minutes is equivalent to 6 days, 22 hours and 40 minutes.
Some bacteria are placed in a small dish. Every minute, the number of bacteria in the dish doubles. After one hour, the dish is completely full of bacteria. After how many minutes was the dish half full of bacteria?
Solution: As the number of bacteria doubles each minute, and the dish is full after 60 minutes, the dish must have been half full after 59 minutes.
On 1st January 2021 Simon writes the number 1 on a board. Every day after this, he divides the previous day’s number by 1 more than the previous day’s number, then writes this new number on the board. What number does he write on 31st December 2021?
Solution: On 2nd Jan, Simon writes 1/2 on the board. On 3rd Jan, he divides 1/2 by 3/2 to obtain 1/3, so writes 1/3 on the board. On 4th Jan he divides 1/3 by 4/3 so writes 1/4 on the board. This pattern continues, with Simon writing 1/n on the nth day of the year. Hence on 31st Dec 2021, the 365th day of the year, he writes 1/365.
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
