Systematic Listing

Take each starter in turn and pair it with every main course.

Start with 2 as the first digit and list every second digit, then do the same for 5 and for 8.

List the pairs of coins in order so that you don't miss one or count one twice.

Group the outcomes by how many heads they have.

Fix the first digit as 1 and list every possible ending, then do the same with 2.

Fix the first digit as 3 and list every possible second digit, then do the same for 7.

1. First digit 3: 31, 34, 36.
2. First digit 7: 71, 74, 76.
3. There are 2\times3=6 codes: 31,\ 34,\ 36,\ 71,\ 74,\ 76.

Read across the row and up the column of the empty box.

1. The empty box is in the row for 5 on spinner A and the column for 4 on spinner B.
2. 5+4=9.
3. The missing total is 9.

Write each pair with the letters in alphabetical order and look for repeats.

1. The fruits must be different, so pairs like AA are not allowed.
2. Pairs with A: AB, AC, AD. Pairs with B (not A): BC, BD. Pairs with C (not A or B): CD.
3. CB is the same pair as BC, so it is a repeat, and CD is missing.
4. There should be 6 different pairs.

Fix the first digit as 4 and list all three possible second digits, then do the same for 6 and for 9.

1. First digit 4: 44, 46, 49.
2. First digit 6: 64, 66, 69.
3. First digit 9: 94, 96, 99.
4. There are 9 numbers.

A number is a multiple of 3 if it is in the 3 times table. Check each number on the list in turn.

1. Go through the list in order: 18=3\times6, 36=3\times12, 63=3\times21, 81=3\times27.
2. The others, such as 13 and 16, leave a remainder when divided by 3.
3. The multiples of 3 are 18,\ 36,\ 63,\ 81.

An odd number must end in an odd digit. Which cards can go last?

1. The number must end in 3 or 9.
2. Ending in 3: 43, 83, 93.
3. Ending in 9: 39, 49, 89.
4. The odd numbers are 39,\ 43,\ 49,\ 83,\ 89,\ 93.

Fix the first card, then list the two ways to place the other two cards.

1. Starting with 2: 267, 276.
2. Starting with 6: 627, 672.
3. Starting with 7: 726, 762.
4. There are 6 numbers.

List all six drink and cake pairs and their totals, then check for totals that appear twice.

1. Tea: 1.20+0.90=2.10 and 1.20+1.20=2.40.
2. Coffee: 1.50+0.90=2.40 and 1.50+1.20=2.70.
3. Juice: 0.80+0.90=1.70 and 0.80+1.20=2.00.
4. £2.40 appears twice, so there are 5 different amounts: £1.70, £2.00, £2.10, £2.40, £2.70.

An ordinary dice shows 1 to 6. For each first score, what would the second score need to be?

1. First 1 needs second 8: not possible. First 2 needs second 7: not possible.
2. First 3 and second 6; first 4 and second 5; first 5 and second 4; first 6 and second 3.
3. The possible first scores are 3,\ 4,\ 5,\ 6.

List all nine pairs in order and work out each product. Some scores come up more than once.

1. With 2: 2\times1=2, 2\times2=4, 2\times4=8.
2. With 3: 3\times1=3, 3\times2=6, 3\times4=12.
3. With 6: 6\times1=6, 6\times2=12, 6\times4=24.
4. 6 and 12 appear twice. The different scores are 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 24.

Fix the first card (it cannot be 0), then place the other two cards both ways.

Part (a)

1. Starting with 5: 507, 570.
2. Starting with 7: 705, 750.
3. Starting with 0 does not give a three-digit number.
4. The numbers are 507,\ 570,\ 705,\ 750.

Part (b)

1. A multiple of 5 ends in 0 or 5.
2. 570, 705 and 750 do. 507 does not.
3. So 3 of the numbers are multiples of 5.

List all twelve two-digit numbers in order first, then check each one for factors.

1. All twelve numbers: 12, 17, 19, 21, 27, 29, 71, 72, 79, 91, 92, 97.
2. Even numbers are not prime: 12, 72, 92. 21=3\times7, 27=3\times9 and 91=7\times13 are not prime.
3. The prime numbers are 17,\ 19,\ 29,\ 71,\ 79,\ 97.

Group the outcomes by how many heads they have.

Part (a)

1. Exactly two heads means two H and one T.
2. The T can be third, second or first: HHT, HTH, THH.
3. HHH has three heads, and HTT has only one.

Part (b)

1. At least two heads means two heads or three heads.
2. That is HHT, HTH, THH and HHH: 4 outcomes out of 8.
3. The probability is \frac{4}{8}=\frac{1}{2}.

Try 1 pen, 2 pens, 3 pens and so on. Is the money left over an exact number of 20p pencils?

1. £2.40 is 240p.
2. 1 pen: 240-30=210p, not a multiple of 20. 2 pens: 180p, which is 9 pencils.
3. 3 pens: 150p, no. 4 pens: 120p, 6 pencils. 5 pens: 90p, no. 6 pens: 60p, 3 pencils. 7 pens: 30p, no.
4. 8 pens cost exactly 240p, leaving no money for a pencil, so that is not allowed.
5. Nia could buy 2, 4 or 6 pens.

List all the orders systematically: start with A first, then B first, then C first.

Part (a)

1. All the orders: ABC, ACB, BAC, BCA, CAB, CBA. That is 6 orders.
2. Amy and Ben are next to each other in ABC, BAC, CAB and CBA.
3. So there are 4 orders.

Part (b)

1. In ACB and BCA, Cal sits between Amy and Ben, so they are apart.
2. That is 2 orders apart and 4 orders together, so Cal is right.