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Simultaneous logic puzzles

Visual algebra grids and balance-scale puzzles.

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In Lesson 5.4 there was one mystery number. Now there are two or three, and one clue on its own is not enough. Put the clues together, like a detective, and every mystery number gives itself away. In this lesson you will learn how to:

  • solve picture puzzles where each shape stands for a number
  • find the clue to start with
  • swap things on a balance scale
  • compare two balance scales
  • crack algebra grids with row and column totals
  • find two numbers from their total and their difference
  • use guess, check and improve
Dark blue: numbers Red: letters, the numbers we are looking for Shapes: each one is a mystery number

1. Picture puzzles

Key idea In a picture puzzle, each shape stands for a number. The same shape is always the same number. Start with a clue that has only one kind of shape in it. Then use the number you found in the next clue.

Example: what are the star and the circle worth?

  1. The first clue has only stars. Three stars make 15:15 ÷ 3 = 5
  2. Put 5 in for the star in the second clue:
  3. 5 + what = 11? Take away 5, as in Lesson 5.4:11 − 5 = 6
  4. Check both clues.5 + 5 + 5 = 15 ✓   5 + 6 = 11 ✓
AnswerThe star is 5 and the circle is 6.

A chain of clues

Two triangles make 8, so a triangle is 4. Then 4 + square = 10, so the square is 6. Then 6 + two circles = 16, so the two circles make 10 and one circle is 5. Each clue gives you the key to the next one.

Watch out Two stars that make 12 means one star is 6, not 12. Count how many times a shape appears, and share the total between them.

2. Balance puzzles: swap it

Key idea A level balance scale is a clue, just like an equation. If one circle balances two stars, they weigh the same. So you can swap a circle for two stars on any scale, and it stays level. Swap until there is only one kind of shape left.

Example: what do the star and the circle weigh?

Scale A
Scale B15 g
  1. Scale A says one circle weighs the same as two stars. Swap the circle on scale B for two stars:
15 g
  1. Now three stars weigh 15 grams:15 ÷ 3 = 5
  2. A star weighs 5 grams, so a circle weighs two stars:5 + 5 = 10
  3. Check scale B.10 + 5 = 15 ✓
AnswerA star weighs 5 g and a circle weighs 10 g.
Watch out After you swap, count all the stars: the new ones and the ones that were there already. Here that is 3 stars, not 2.

3. Balance puzzles: spot the difference

Key idea When two scales have the same things on them, except that one has some extras, the extras must weigh the difference between the two scales.

Example: what do the star and the circle weigh?

Scale A16 g
Scale B24 g
  1. Scale B has the same circle and star as scale A, plus two extra stars.
  2. Scale B weighs 8 grams more:24 − 16 = 8
  3. So the two extra stars weigh 8 grams, and one star weighs:8 ÷ 2 = 4
  4. Use scale A: circle + 4 = 16, so the circle weighs:16 − 4 = 12
  5. Check scale B.12 + 4 + 4 + 4 = 24 ✓
AnswerA star weighs 4 g and a circle weighs 12 g.

The same idea in a shop

3 apples and a pear cost 65p. 2 apples and a pear cost 50p. The first basket has one extra apple, and it costs 15p more, so an apple costs 15p. Then 2 apples cost 30p, and 50 − 30 = 20, so the pear costs 20p. Check: 45 + 20 = 65 ✓

Watch out The difference is what all the extras weigh together. If there are two extra stars, share the difference between them.

4. Algebra grids

Key idea In an algebra grid, the shapes in each row add up to the number at the end of the row, and the shapes in each column add up to the number underneath. Find a row or column with only one kind of shape you do not know yet. Solve it, then look for the next one.

Example: find each shape, then the missing total

  1. Row 1 is all stars. Three stars make 12:star = 12 ÷ 3 = 4
  2. Row 2 is a star and two circles. Take away the star: 18 − 4 = 14, and share by 2:circle = 7
  3. Row 3: a triangle, a circle and a star. 7 + 4 = 11, and 20 − 11 = 9:triangle = 9
  4. The missing total is column 3: star, circle, star.4 + 7 + 4 = 15
  5. Check column 1: star, star, triangle.4 + 4 + 9 = 17 ✓
AnswerStar 4, circle 7, triangle 9. The missing total is 15.
Watch out Read each row across and each column down. Every shape counts once for each time it appears, so a row with two circles has the circle’s number in it twice.

5. Two numbers from two clues

Key idea “Two numbers add up to 20, and their difference is 4.” Draw a bar for each number. The bigger bar is the smaller bar plus 4. Take the 4 away, and what is left is two equal bars. Share it, and you have the smaller number.

Example: two numbers add up to 20. Their difference is 4. What are they?

big?4small?20
  1. Take the difference off the total:20 − 4 = 16
  2. That is two equal bars. Share by 2 for the smaller number:16 ÷ 2 = 8
  3. Add the difference back for the bigger number:8 + 4 = 12
  4. Check both clues.8 + 12 = 20 ✓   12 − 8 = 4 ✓
AnswerThe numbers are 8 and 12.

One is three times the other

big???small?24

Two numbers add up to 24, and one is 3 times the other. That is 4 equal boxes, so one box is 24 ÷ 4 = 6, and the bigger number is 3 × 6 = 18.

With letters (Year 6)

The first example with letters is a + b = 20 and a − b = 4. It is the same puzzle: b = 8 and a = 12.

2a + b = 17 and a + b = 11: the first clue has one extra a, and 17 − 11 = 6. So a = 6, and b = 11 − 6 = 5. It is the spot-the-difference idea from section 3.

Watch out Halving the total gives 10 and 10, which have no difference at all. Take the difference away before you share.

6. Guess, check and improve

Key idea Some puzzles have two kinds of thing and two totals: “A farm has hens and sheep. There are 10 heads and 28 legs.” Make a guess, check it against both totals, then improve it. Write each try in a table so you can see which way to go.

Example: 10 heads and 28 legs. How many hens and how many sheep?

HensSheepHeadsLegs
10010208 too few
731014 + 12 = 262 too few
641012 + 16 = 28✓

A quicker way: start with all hens, which is 20 legs. That is 8 too few. Each time a sheep takes the place of a hen, there are 2 more legs. 8 ÷ 2 = 4, so 4 sheep and 6 hens.

Watch out Check both totals. 7 sheep and 0 hens have 28 legs, but only 7 heads. And answer what was asked: the number of hens and sheep, not the number of legs.

The mistakes that cost marks

  1. Starting with the wrong clue. Guessing in a clue that has two shapes you do not know. start with a clue that has only one kind of shape you do not know.
  2. Forgetting that a shape appears more than once. Saying a star is 12 when two stars make 12. share the total: one star is 12 ÷ 2 = 6.
  3. Missing a swap. Swapping one circle on a scale but not the other. swap every one, then count all the shapes that are left.
  4. Giving the difference as the answer. Saying a star weighs 8 g when two extra stars weigh 8 g. the difference is for all the extras together: 8 ÷ 2 = 4 g.
  5. Halving the total straight away. Saying 10 and 10 add up to 20 with a difference of 4. take the difference off first: 20 − 4 = 16, then 8 and 12.
  6. Stopping after one number. Finding the star but not the circle. put the number you found back into a clue to get the next one.
  7. Checking only one clue. 7 sheep have 28 legs, but only 7 heads. check your numbers in every clue.
  8. Answering the wrong thing. Writing the number of legs when the question asks for sheep. read the question again before you write the answer.

Quick checklist

Tick each one when you can do it without help.

Download

Print these and work through them away from the screen.

Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

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