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?
Solutions: 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 positive numbers are factors of both 45 and 60?
Solution: The common factors of 45 and 60 are precisely the factors of 15, namely 1, 3, 5 and 15. So the answer is 4.
How many positive numbers are factors of 100 but not 50?
Solutions: We require those factors of 100 that are divisible by 4, namely 4, 20 and 100, So the answer is 3
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 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 tall
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?
Answer: It is not possible
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.
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?
Solutions: 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.
Two fifths of a number is 5. Find two thirds of the number.
Answer: 25/3 or 8 1/3
Solution: One fifth of the number is 2.5, so the number is 5×2.5 = 12.5. Finally, two thirds of 12.5 is 2×12.5/3 = 25/3 or 8 1/3
A two-pence piece is glued to a table, with Heads facing up. Another two-pence piece is placed next to it, so that the two coins are touching. Then the second coin rolls around the first (fixed) coin without slipping, until it returns to its starting position. Through what angle does the second coin turn?
Answer: 720 degrees
Solution: The second coin makes two complete revolutions (try it for yourself!), so it turns through 2×360 = 720 degrees
Daria thinks of a number. She doubles it, then subtracts 7, then halves the result, ending up with -2. What number did she start with?
Solution: Daria starts with 1.5. She doubles it to get 3, subtracts 7 to get -4, then halves this to end up with -2.
Samuel thinks of a number. He cubes it, then adds 30, then multiplies the result by -1, ending up with the number he first thought of. What is this number?
Solution: Samuel starts with -3. He cubes it to get -27, adds 30 to get 3, then multiplies this by -1 to get back to -3
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
A fish weighs 3kg plus a third of its weight. How much does the fish weigh?
Solution: One third plus two thirds makes a whole, so 3kg is two thirds of the weight. Hence 1.5kg is one third of the weight, and the total weight of the fish is 1.5×3=4.5kg
In an isosceles triangle, one of the angles is 40 degrees, and one of the other angles is x degrees. What are the possible values of x?
Answer: 40, 70 and 100
Solution: If 40 degrees is one of the two base angles, then the other two angles are 40 degrees and 100 degrees. If, however, 40 degrees is the angle at the apex, then the two base angles are equal to 70 degrees. So the possible values of x are 40, 70 and 100
In how many ways can 60 be written as a sum of consecutive positive odd numbers?
Solution: Two (5+7+9+11+13+15 and 29+31).
Suppose all the arrangements of the letters in the word FACTOR are listed in dictionary order (so the list begins 1 ACFORT, 2 ACFOTR, 3 ACFROT, …). In what position does the word FACTOR appear?
Answer: the 245thposition
Solution: There are 6x5x4x3x2x1=720 arrangements of the letters in the word FACTOR. The first 120 of them begin with an A; the next 120 begin with a C. Then we have 241 FACORT, 242 FACOTR, 243 FACROT, 244 FACRTO, and 245 FACTOR. So FACTOR appears in the 245th
Monica cuts a cake into 10 large pieces. She then takes half of these and cuts them into 3 medium pieces each. Finally, she takes a third of the medium pieces and cuts each of them into 2 small pieces.
(i) How many pieces of cake are there at the end?
Solution: Of the 10 large pieces, 5 are left alone. The other 5 are cut into 15 medium pieces. Of these 15 pieces, 10 are left alone. The other 5 are cut into 10 small pieces. So at the end, there are 5 large, 10 medium and 10 small slices; that is 25 pieces in total.
(ii) Josh takes one piece of each size. What fraction of the original cake does this represent?
Solution: Each large piece represents 1/10 of the original cake. Each medium piece represents 1/30 of the original cake. Each small piece represents 1/60 of the original cake. So the fraction represented by Josh’s portion is (1/10)+(1/30)+(1/60)=(6/60)+(2/60)+(1/60)=(10/60)=1/6 (One sixth).
Amelia is three times as old as Jay. In four years, Amelia will be twice as old as Jay. How many years older than Jay is Amelia?
Solution: 8 years – Amelia is currently 12, Jay is currently 4
There is a circular table with six empty seats around it. Charlie and Dilshan pick different seats at random. What is the probability they sit opposite each other?
Solution: Wherever Charlie sits, there will be five seats remaining for Dilshan, only one of which is opposite Charlie. So the answer is 1/5
How many primes less than 200 contain a digit 0?
Solution: 4 (101, 103, 107, 109)
A rectangular grid of squares is coloured blue. A border, one square wide, is constructed around the rectangular grid, and painted yellow. The number of blue squares is equal to the number of yellow squares. Another rectangular grid of squares with different dimensions is also coloured blue, and a similar yellow border is constructed. The number of blue squares in this grid is equal to the number of yellow squares comprising its border. How many squares are used in total in both grids combined?
Solution: 108 (One grid has dimensions of 3×10, the other 4×6)
Annie thinks of a number. She triples it, then subtracts 10, ending up with -25. What was her original number?
Solution: Reverse the operations Annie carried out: add 10 to -25 to obtain -15, then divided -15 by 3 to obtain -5
What is the smallest number divisible by exactly one prime number and exactly six other positive integers?
Solution: Any such number will be of the form p x p x p x p x p x p for some prime number p. The smallest number of this type is 2x2x2x2x2x2=64 (Its factors are 1, 2, 4, 8, 16, 32 and 64)
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.