# Heads and Legs, Guess and Check

> Solve "two kinds of things" puzzles by guessing smartly or by pretending they are all one kind, plus sum and difference problems.

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

### Story time: The farm visit
On a farm visit, Chloe looks over a fence at some ducks and rabbits. She counts 16 heads and 54 legs. "How many rabbits?" asks the farmer with a grin.

Chloe imagines all 16 animals are ducks. That would be $16 \times 2 = 32$ legs. But there are 54 legs, so 22 legs are missing. Every time she swaps a duck for a rabbit, she adds 2 legs. So she needs $22 \div 2 = 11$ swaps. There are 11 rabbits and 5 ducks!

The farmer is impressed. Chloe used one of the most useful tricks in problem solving: **pretend they are all the same, then fix it**.

**In short:** Guess and check means making a guess, checking it, and using what you learn to make a better one. For heads-and-legs problems, pretend all the animals are one kind, see how far off the total is, then swap some to fix it.

## Before you start

- Multiplying and dividing whole numbers.
- Money: 100 cents $= 1$ dollar.
- Simple equations can help, but they are not needed here.

## Method 1: guess, check, improve

Make a sensible guess. Check it. Use what went wrong to make a better guess.

### Worked example: Ducks and rabbits by guessing
16 heads, 54 legs.

| Rabbits | Ducks | Legs | Check |
|---|---|---|---|
| 8 | 8 | $32 + 16 = 48$ | too few |
| 10 | 6 | $40 + 12 = 52$ | still too few |
| 11 | 5 | $44 + 10 = 54$ | correct |

### Exam tip: Put your guesses in a table
A table stops you repeating a guess and shows you which way to move.

## Method 2: pretend they are all one kind

### Key idea: All the same, then swap
1. Pretend every item is the **cheaper** (or smaller) kind.
2. Work out the total that would give.
3. Find how far off you are.
4. Each swap to the other kind changes the total by the **difference** between the two kinds.
5. Number of swaps $=$ how far off $\div$ difference per swap.

### Worked example: Girls and dogs
At the park there are 24 heads and 62 feet. How many girls and how many dogs?

All girls: $24 \times 2 = 48$ feet. Missing: $62 - 48 = 14$ feet. Each dog adds 2 more feet than a girl. Dogs: $14 \div 2 = 7$. Girls: $24 - 7 = 17$.

### Worked example: Lollies and money
Lollipops cost 25 cents and Lifesavers cost 30 cents. Stacey spent 2.35 dollars on 9 items.

All lollipops: $9 \times 25 = 225$ cents. Missing: $235 - 225 = 10$ cents. Each swap adds 5 cents. Lifesavers: 2. Lollipops: 7.

### Worked example: A board game
Throwing an even number moves you 5 spaces forward. Throwing an odd number moves you 3 spaces back. After 11 throws you are 31 spaces forward. How many evens did you throw?

All evens: $11 \times 5 = 55$ forward. That is $55 - 31 = 24$ too many. Swapping an even for an odd changes your total by $5 + 3 = 8$. Odds: $24 \div 8 = 3$. Evens: 8.

### Watch out: Size of each swap
When one kind moves you forward and the other back, a swap changes the total by **both** amounts added, like $5 + 3 = 8$ above, not by $5 - 3$.

## Sum and difference

"Two numbers add to 67 and their difference is 11."

### Key idea: Halve it
Take away the difference and halve: smaller $= \frac{67 - 11}{2} = 28$. The larger is $28 + 11 = 39$.

### Challenge: Selective level: three kinds of stamps
A man spends exactly 2 dollars on 5c, 1c and 10c stamps. He buys 10 times as many 1c stamps as 5c stamps, and half as many 10c stamps as 5c stamps. How many 5c stamps?

Think in bundles. One bundle: 2 five-cent stamps, 20 one-cent stamps and 1 ten-cent stamp. Its value is $10 + 20 + 10 = 40$ cents. 200 cents is 5 bundles, so there are $5 \times 2 = 10$ five-cent stamps.

Mr Wood makes tables with 4 legs and stools with 3 legs. Nine customers each order 8 items, and no two orders are the same. The only possible orders are 0 tables and 8 stools, 1 and 7, and so on up to 8 tables and 0 stools. That is exactly 9 orders. Tables: $0 + 1 + \dots + 8 = 36$, stools: also 36. Legs: $36 \times 4 + 36 \times 3 = 252$.

**Mind map**

```text
Two kinds of things
  Guess and check
    Make a table
    Improve each guess
  Pretend all one kind
    Find how far off
    Divide by the change per swap
  Forward and back
    Swap changes by both amounts
  Sum and difference
    (sum - difference) ÷ 2 = smaller
```

## Quick summary

- Guess, check, improve: use a table to steer your guesses.
- Pretend everything is one kind, find how far off you are, then divide by how much each swap changes the total.
- When one kind adds and the other takes away, each swap changes the total by both amounts together.
- Sum and difference: smaller $=$ (sum $-$ difference) $\div 2$.
- Always check your answer in the original story.

## Quiz

Try these questions from the chapter quiz.

1. Two numbers add to 67 and their difference is 11. What is the smaller number?
   - A. 11
   - B. 28
   - C. 33.5
   - D. 39
   - Answer: B. 28. $\frac{67 - 11}{2} = 28$.

2. There are 16 heads and 54 legs among ducks and rabbits. How many rabbits?
   - A. 5
   - B. 11
   - C. 8
   - D. 13
   - Answer: B. 11. All ducks: 32 legs. 22 legs missing, 2 per rabbit: 11 rabbits.

3. A farm has chickens and cows: 10 heads and 28 legs. How many cows?
   - A. 4
   - B. 6
   - C. 5
   - D. 8
   - Answer: A. 4. All chickens: 20 legs. 8 missing, 2 per cow: 4 cows.

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