# Logic Puzzles

> Race orders, digit clues, letter sums, remainders and working backwards, solved with a clear plan.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/logic-puzzles/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The swimming carnival mystery
After the 400 m freestyle, the results board breaks. The timekeeper only remembers some clues: "Michael finished 2 m ahead of Barry. Scott beat Alex. Alex was 2 m ahead of Michael. Stuart came second last."

Ana, the team captain, draws a line and puts names on it. Alex is in front of Michael, who is in front of Barry. Scott is in front of Alex. With 5 swimmers, second last means 4th, so Stuart goes there. The order must be Scott, Alex, Michael, Stuart, Barry.

Logic puzzles need no hard maths. They need patience, a diagram, and clues used one at a time.

**In short:** Logic puzzles are solved by using the clues one at a time. Draw a diagram, list the possible cases and test each one, work backwards when you know the end, and check your answer against every clue.

## Before you start

- Odd and even numbers, multiples and remainders.
- Adding columns with carrying.
- Working backwards (see Solving Equations).

## Ordering puzzles

### Key idea: Draw a line
Draw a number line or a race track. Place the person you know most about first. Then add each clue. Leave gaps for people who are not placed yet.

### Worked example: The cross-country race
Lloyd finished 2 m ahead of Christian. Hamish finished 4 m ahead of Stuart. Stuart was 1 m ahead of Christian. Paul came second.

Put Christian at 0 m. Then Stuart is at 1 m, Lloyd at 2 m and Hamish at $1 + 4 = 5$ m. Paul is second, so he goes just behind Hamish.

Order: Hamish, Paul, Lloyd, Stuart, Christian.

## Digit clue puzzles

### Worked example: The safe
A safe's code is a 3-digit number. The digits add to 14. Two digits are odd. The tens digit is the largest. The tens digit is twice the units digit.

The tens digit is twice the units, so try the pairs 2 and 1, 4 and 2, 6 and 3, 8 and 4.

- Tens 8, units 4: hundreds $= 14 - 12 = 2$. No odd digits. No.
- Tens 6, units 3: hundreds $= 14 - 9 = 5$. Digits 5, 6, 3: two odd, 6 is largest. Yes!
- Tens 4, units 2: hundreds $= 8$, bigger than the tens. No.

The code is 563.

### Exam tip: Make a list of cases
When there are only a few possibilities, list them all and cross out the ones that break a clue.

## Letter sums (cryptarithms)

Each letter stands for a different digit.

### Worked example: A + A + A = BA
Three A's add to a number ending in A. $3 \times A$ must end in A. Try digits: $3 \times 5 = 15$ ends in 5. So $A = 5$ and $B = 1$.

### Worked example: MY + Y = YM
$10M + Y + Y = 10Y + M$, so $9M = 8Y$. The only digits that work are $M = 8$, $Y = 9$: $89 + 9 = 98$.

### Worked example: CCC + AAA + KKK = EEE
Each column is $C + A + K = E$, with no carrying. Three different non-zero digits add to at least $1 + 2 + 3 = 6$. So of 333, 444, 555 and 666, only **666** is possible.

## Remainder puzzles

### Worked example: The apple box
A box holds at most 40 apples. Taking them out 2, 3, 5 or 6 at a time always leaves 1. Taking them out 7 at a time leaves 3. How many apples?

One less than the number must be a multiple of 2, 3, 5 and 6, which means a multiple of 30. So the number is 1, 31 or 61. Only 31 is at most 40 and above 1, and $31 = 4 \times 7 + 3$. Answer: 31.

## Working backwards

"I'm thinking of a number. I multiply it by 15 and add 36 to get 141." Undo in reverse: $141 - 36 = 105$, then $105 \div 15 = 7$.

"Kristy spent 36 dollars and 43 dollars and had 32 dollars left." She started with $36 + 43 + 32 = 111$ dollars.

### Challenge: Selective level: the coin swap
24 one-cent coins are in a row. Then every 2nd coin is replaced by a 2c coin, every 3rd by a 5c coin, every 4th by a 10c coin, every 5th by a 20c coin, every 6th by a 50c coin and every 7th by a 1 dollar coin. Later swaps replace earlier ones. What is the total?

Each position ends with the coin from the **largest** of 2 to 7 that divides it.

- 1, 11, 13, 17, 19, 23: still 1c (6 coins, 6c)
- 2, 22: 2c (4c)
- 3, 9: 5c (10c)
- 4, 8, 16: 10c (30c)
- 5, 10, 15, 20: 20c (80c)
- 6, 12, 18, 24: 50c (200c)
- 7, 14, 21: 1 dollar (300c)

Total: $6 + 4 + 10 + 30 + 80 + 200 + 300 = 630$ cents $= 6.30$ dollars.

**Mind map**

```text
Logic puzzles
  Ordering
    Draw a line
    Place one clue at a time
  Digit clues
    List the cases
    Cross out what fails
  Letter sums
    Look at the last column
    Check for carrying
  Remainders
    One less is a common multiple
  Working backwards
    Undo in reverse order
```

## Quick summary

- Draw a diagram for ordering puzzles and add one clue at a time.
- When few cases are possible, list them all and test each one.
- In letter sums, start with the units column and think about carrying.
- If a number leaves remainder 1 for several divisors, one less than it is a common multiple.
- Work backwards by undoing each step in reverse.
- Always check your answer against every clue.

## Quiz

Try these questions from the chapter quiz.

1. I multiply a number by 6 and subtract 7 to get 59. What is the number?
   - A. 9
   - B. 11
   - C. 10
   - D. 12
   - Answer: B. 11. Undo: $59 + 7 = 66$, then $66 \div 6 = 11$.

2. Kristy spent 36 dollars on a skateboard and 43 dollars on a jacket. She had 32 dollars left. How much did she start with?
   - A. 79 dollars
   - B. 111 dollars
   - C. 101 dollars
   - D. 75 dollars
   - Answer: B. 111 dollars. $36 + 43 + 32 = 111$.

3. I multiply a number by 15 and add 36 to get 141. What is the number?
   - A. 7
   - B. 9
   - C. 11
   - D. 105
   - Answer: A. 7. $141 - 36 = 105$ and $105 \div 15 = 7$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/logic-puzzles/#quiz