(Please note some solutions are dependent on images and are omitted here. If you need a

particular solution please e-mail).

**Volume ****10**

**CIRCA 28: Solutions**

**Front cover/A medieval mix**:

Which net?

Ethel is holding the correct net.

How many triangles?

The tray on the left contains 16 triangle shapes.

The tray on the right contains 20 triangle shapes.

Impossible ‘goings-on’s

Some of the impossible things are: the ladder, frames of cube and pyramid, chair, pillars,

inside-out stairs, the three pipes, silhouettes in the trees and odd castellations.

**Pages 2 and 3/You can count on your fingers**

The Venerable Bede was the first to record, in a systematic way, a finger reckoning system in

the 7th century.

Egyptian man

He is showing 200 (the Ancient Egyptians used their left hand for numbers over 100).

Roman counter: This could be confused with 3. The folded down fingers are further down the

palm for 9.

Italian merchants

The man wants 600 florins. The woman is offering 550.

Thomas at the fair: 56 pence.

Hands

The hands show the multiplication 8 x 9. The product is 72 (70 + 2 x 1).

More multiplying

All the multiplications shown can easily be done on your hands.

Multiplying 7 x 16

To multiply 7 x 16, Thomas would have split the calculation. Though he would probably know

that 7 x 10 was 70, he could have done it using his hands. Then he would have done 7 x 6 and

added the two products together. There are various other ways it could be done such as 7 x 8,

then double the answer.

**Pages 4 and 5/****Knights of the Triangle Table**

(1) Scalene (2) Isosceles (3) Not a triangle (4) Equilateral

(5) Acute (6) Not a triangle (7) Obtuse (8) Right-angled

Shields

A (5) B (6) C (3) D (1) E (4) F (2)

**Page 6 and 7/****How many degrees?**

Completed table:

Shape Number Sum of angles

of sides (degrees)

Triangle 3 180°

Quadrilateral 4 2 x 180° = 360°

Pentagon 5 3 x 180° = 540°

Hexagon 6 4 x 180° = 720°

Septagon 7 5 x 180° = 900°

Octagon 8 6 x 180° = 1080°

The number of triangles (180°) is 2 less than the number of sides.

Diners’ flags

Edgar (540°) (C)

Rosamund ( 900°) (B)

Maude (1440°) (F)

Harold (360°) (E)

Roland (1260°) (D)

Alyce (720°) (A)

Irene’s flag

The sum of the degrees in Irene’s 12-sided flag is 1800° .

**Page 8 and 9****/Coordinates in the castle**

Martha the maiden walks an irregular quadrilateral.

John the jester walks a square.

Friar Doug walks a trapezium.

Graham the guard walks a rectangle.

Sarah the cook walks a parallelogram.

A rhombus is the shape which is not walked.

If Graham the guard pauses at (8, 7) he could meet Sarah the cook.

**Pages 10 and 11****/Use your head**

Paper cuts and folds

1 (B) 2 (R) 3 (A) 4 (I) 5 (N) 6 (P) 7 (O) 8 (W) 9 (E) 10 (R)

Folding cubes

BOXES

**Pages 12 and 13****/Folding geometry**

No solutions

**Pages 14 and 15****/Are you puzzled?**

Knight match

(1) &Mac221; (E) (2) &Mac221; (A) (3) &Mac221; (I) (4) &Mac221; (B) (5) &Mac221; (C)

(6) &Mac221; (G) (7) &Mac221; (H) (8) &Mac221; (D) (9) &Mac221; (F)

1. Step up

Clive has to climb 6 steps to visit Fred.

From the information given you can work out that:

D(awn) lives at the bottom of the hill.

B(renda) lives between D and C(live).

A(nne) lives at top of the hill because F(red) has to climb to visit A and E(llen) has to climb to visit A.

Fred is of course “me”.

2. Piles of plates

There are 30 plates.

3. In the right order

7, then 8 (8 x 2 = 16), then 9 (9 x 3 = 27).

7 + 16 + 27 = 50.

4. Pie buy

You could get 7 pies.

£16 + £8 + £4 + £2 + £1 + 50p + 25 p = £31.75

5. Rectangle count

There are 18 rectangles in the design on Sir Quadril’s shield.

6. Match up

Rob’s number is 86 (98).

Dob’s number is 66 (66 ÷ 2 = 33, 66 x 2 = 132, 132 - 33 = 99)

Bob’s number is 45 (54)

7. Cover up

The five ‘L-tiles’ can be laid without any overlapping or any hanging off the board so that

any cell on the grid can be left uncovered.

**Page 16 (back cover)****/Whose Buyseye?**

Brad got the bullseye. Here’s why:

The total score for both cousins was 160 points (2 x 5, 5 x 10, 2 x 25 and 1 x 50).

Which meant that both scored 80.

Alice scored 45 with her first 3 arrows, so no bullseye. She scored 35 with her remaining 2 arrows (there were no misses) neither of which could be a bullseye. TOP.

**CIRCA 29: Solutions**

**Front cover/Mrs Bloomer's teapot**:

If Mrs Bloomer glued the bits together she would have a whole teapot.

Each of the three fractions found are a third or equivalent to a third.

3/9 is a third, two-sixths is a third and a third.

**Pages 2 and 3/Be a code-cracker**

Substitution code

THE MIDDLE NAME OF WINSTON CHURCHILL WAS LEONARD

Top secret

Message 1

TO U BOAT ZERO ONE COLLECT FISH FOR SUPPER AT THREE COMMA THREE

(move 2 code e.g. C = A, B = Z)

Message 2

TO U BOAT ZERO TWO COOL DOWN AT TWO COMMA EIGHT

(move 7 code e.g. H = A, G = Z)

Message 3

TO U BOAT ZERO THREE CREW NEED TO SHOWER PROCEED TO FIVE COMMA SEVEN

(move 4 code e.g. E = A, D = Z)

U-Boat 01 goes to G – the row boat at (3,3)

U-Boat 02 goes to A – the iceberg at (2,8)

U-Boat 03 goes to D – the whale at (5,7)

The use of the word ‘code’ is not strictly correct. What most of us think of as a code is in fact a cipher. A code is a word or phrase with a prearranged meaning e.g. ‘the moon is full’ would signal that the army was ready for battle.

**Pages 4 and 5/****Fraction rations**

1. 1 minute

2. 16 coupons

3. Less (3 coupons)

4. 25 pieces

5. 4.55

6. One third

7. 3 pairs

8. 2 coupons

The letters read BLACKOUT

**Page 6 and 7/****Cloudy fractions**

1. more than 9/10

2. 3/10

3. less than 1/10

4. 8/10

5. 5/10

6. 6/10

In order, the fractions give the secret word 'cloudy'.

**Page 8 and 9****/Follow the fractions**

Florence (1/2): 2/4, 7/14, 5/10, 3/6, 4/8, 6/12 Apple Farm

Percy (3/5): 6/10, 12/20, 15/25, 30/50, 18/30, 9/15 The Baker

Ethel (3/4): 6/8, 21/28, 15/20, 9/12, 12/16, 18/24 Post Office

Harry (2/3): 4/6, 12/18, 6/9, 8/12, 10/15, 14/21 The Butchers

**Pages 10 and 11****/Wakey, wakey!**

Paper cuts and folds

1. S 2. P 3. U 4. D 5. B 6. A 7. S 8. H 9. E 10. R

which spells out SPUDBASHER.

**Pages 12 and 13****/Canteen calculations**

1. 38 pieces (quarters) of cake – Y

2. one dozen (12) mugs of beef teas – R

3. 12 big pots of tea – O

4. 57 women (171 men) – I

5. (32 rhubarb) 80 gooseberry – C

6. 480 mugs of tea – T

7. One third of the day – V

The solution letters can be rearranged to spell ‘VICTORY’.

**Pages 14 and 15****/Are you puzzled?**

Jack’s marbles

Jack has 20 marbles.

(Alan has 24, Ian has 22, Billy has 18 and Charlie has 16.

Plum pudding

The pudding weighs 500g.

The ingredients other than the honey sum to 95% of weight.

The honey therefore equals 5% of the pudding’s weight.

If 5% is 25g, then 100% is 500g.

Egg share

Mr Gibbons should visit Mrs Cummings first. This will ensure that for any number of eggs

over 20 he will retain 5 more eggs than if he visited the air-raid warden first. If he has less

than 20 eggs he must visit Mrs Cummings first or he will not be able to give her 10 eggs.

Sausage sell-out

Mr Bacon had 8 sausages at noon.

(4 + 1 to Mrs Bracknell and 3 sausages left for Mr and Mrs Bacon’s tea.)

Sid’s leave

Sid’s leave is for 48 hours.

**Page 16 (back cover)****/Showtime Sum**

Yes. The show had made enough (excluding any expenses, etc).

Alice put out 12 rows of 20 chairs. That’s 240 seats.

Tickets cost £5.50 each (£22.00 ÷ 4). All three shows were sell outs,

so 720 tickets were sold bringing in £3960. TOP.

**CIRCA 30: Solutions **

**Front cover/Balancing act**:

Only two creatures need to swop places, the snake with the bird.

Each side of the log will then be balanced at 12.5 kg.

**Pages 2 and 3/Numbers from India**

Re-arranging four digits

There are 18 different ways to arrange the digits 8530 to make 4-digit numbers.

If the zero had been 2, 4, 6, 7 or 9 there would have been a further six ways.

8530 5830 3850

8503 5803 3805

8053 5083 3085

8350 5380 3580

8305 5308 3508

8035 5038 3058

Multiplying the ‘gomutrika’ way

(C) and (F) have the wrong products.

The correct products are: (C) 748 and (F) 58280.

**Pages 4 and 5/****A doubling or halving day out**

The answers are: 1. (c), 2. (f), 3. (h), 4. (l), 5. (n), 6. (r), 7. (s), 8. (x), 9. (z).

Using the ‘Solutions Checker’ the word ‘bungalows’ is spelt out. Other Indian words

borrowed/adopted into English include: bangle, chintz, cot, dungaree, juggernaut, jungle,

loot, pyjamas, shampoo, thug.

**Page 6 and 7/****Modest or massive**

To count the grains in just one sack, at the rate of one grain per second, it would take

Shashi 196 days (this assumes counting day and night and no toilet breaks for

more than six months).

**Page 8 and 9****/Operation jungle**

From start (1 + 4) x 4 ÷ 2 + 40 - 1 = 49

From ruins (49 ÷ 7) x 6 ÷ 3 + 4 - 3 = 15

From pool (15 x 2) x 3 ÷ 9 + 50 - 10 = 50

From huts (50 - 5) x 20 ÷ 10 + 15 - 5 = 100 (Finish)

**Pages 10 and 11****/Tuk Tuk time**

1. 30 mins &Mac221; A

2. 16:40 &Mac221; B

3. false &Mac221; E

4. yes &Mac221; N

5. 8 km &Mac221; L

6. 3 km &Mac221; G

When rearranged the letters of the solutions spell out BANGLE.

This is one of many words borrowed from India (see solution to ‘A doubling or halving day’).

**Pages 12 and 13****/Kolam designs**

The solutions for this article need diagrams. Please email.

**Pages 14****/Money in India**

Finding rupees

The solutions for this article need diagrams. Please email.

Changing money.

The answers are: No (G), Yes (A), No (N), Yes (G), No (E), Yes (S) and spells out GANGES.

**Pages 15****/Are you puzzled?**

Tiger hunt

The correct silhouette is (B).

Moving logs

The solutions for this article need diagrams. Please email.

Knotty snake

When stretched out the snake won’t have any knots.

Monkey puzzle

(A) belongs to Hari, (B) belongs to Mitwas, (C) belongs to Aish and (D) belongs to Jaz.

**Page 16 (back cover)****/Mela maths**

Alice and Vishal should buy from ‘Indian Snacks’. This will give them the most samosas for the money they have (£3.00 between them).

At ‘Indian Snacks’ the £3.00 will buy them seven samosas.

At ‘Spicy Eats’ the £3.00 will buy them only six samosas

with 20p left. TOP.

particular solution please e-mail).

Which net?

Ethel is holding the correct net.

How many triangles?

The tray on the left contains 16 triangle shapes.

The tray on the right contains 20 triangle shapes.

Impossible ‘goings-on’s

Some of the impossible things are: the ladder, frames of cube and pyramid, chair, pillars,

inside-out stairs, the three pipes, silhouettes in the trees and odd castellations.

the 7th century.

Egyptian man

He is showing 200 (the Ancient Egyptians used their left hand for numbers over 100).

Roman counter: This could be confused with 3. The folded down fingers are further down the

palm for 9.

Italian merchants

The man wants 600 florins. The woman is offering 550.

Thomas at the fair: 56 pence.

Hands

The hands show the multiplication 8 x 9. The product is 72 (70 + 2 x 1).

More multiplying

All the multiplications shown can easily be done on your hands.

Multiplying 7 x 16

To multiply 7 x 16, Thomas would have split the calculation. Though he would probably know

that 7 x 10 was 70, he could have done it using his hands. Then he would have done 7 x 6 and

added the two products together. There are various other ways it could be done such as 7 x 8,

then double the answer.

(1) Scalene (2) Isosceles (3) Not a triangle (4) Equilateral

(5) Acute (6) Not a triangle (7) Obtuse (8) Right-angled

Shields

A (5) B (6) C (3) D (1) E (4) F (2)

Completed table:

Shape Number Sum of angles

of sides (degrees)

Triangle 3 180°

Quadrilateral 4 2 x 180° = 360°

Pentagon 5 3 x 180° = 540°

Hexagon 6 4 x 180° = 720°

Septagon 7 5 x 180° = 900°

Octagon 8 6 x 180° = 1080°

The number of triangles (180°) is 2 less than the number of sides.

Diners’ flags

Edgar (540°) (C)

Rosamund ( 900°) (B)

Maude (1440°) (F)

Harold (360°) (E)

Roland (1260°) (D)

Alyce (720°) (A)

Irene’s flag

The sum of the degrees in Irene’s 12-sided flag is 1800° .

Martha the maiden walks an irregular quadrilateral.

John the jester walks a square.

Friar Doug walks a trapezium.

Graham the guard walks a rectangle.

Sarah the cook walks a parallelogram.

A rhombus is the shape which is not walked.

If Graham the guard pauses at (8, 7) he could meet Sarah the cook.

1 (B) 2 (R) 3 (A) 4 (I) 5 (N) 6 (P) 7 (O) 8 (W) 9 (E) 10 (R)

Folding cubes

BOXES

(1) &Mac221; (E) (2) &Mac221; (A) (3) &Mac221; (I) (4) &Mac221; (B) (5) &Mac221; (C)

(6) &Mac221; (G) (7) &Mac221; (H) (8) &Mac221; (D) (9) &Mac221; (F)

Clive has to climb 6 steps to visit Fred.

From the information given you can work out that:

D(awn) lives at the bottom of the hill.

B(renda) lives between D and C(live).

A(nne) lives at top of the hill because F(red) has to climb to visit A and E(llen) has to climb to visit A.

Fred is of course “me”.

2. Piles of plates

There are 30 plates.

3. In the right order

7, then 8 (8 x 2 = 16), then 9 (9 x 3 = 27).

7 + 16 + 27 = 50.

4. Pie buy

You could get 7 pies.

£16 + £8 + £4 + £2 + £1 + 50p + 25 p = £31.75

5. Rectangle count

There are 18 rectangles in the design on Sir Quadril’s shield.

6. Match up

Rob’s number is 86 (98).

Dob’s number is 66 (66 ÷ 2 = 33, 66 x 2 = 132, 132 - 33 = 99)

Bob’s number is 45 (54)

7. Cover up

The five ‘L-tiles’ can be laid without any overlapping or any hanging off the board so that

any cell on the grid can be left uncovered.

The total score for both cousins was 160 points (2 x 5, 5 x 10, 2 x 25 and 1 x 50).

Which meant that both scored 80.

Alice scored 45 with her first 3 arrows, so no bullseye. She scored 35 with her remaining 2 arrows (there were no misses) neither of which could be a bullseye. TOP.

If Mrs Bloomer glued the bits together she would have a whole teapot.

Each of the three fractions found are a third or equivalent to a third.

3/9 is a third, two-sixths is a third and a third.

THE MIDDLE NAME OF WINSTON CHURCHILL WAS LEONARD

Top secret

Message 1

TO U BOAT ZERO ONE COLLECT FISH FOR SUPPER AT THREE COMMA THREE

(move 2 code e.g. C = A, B = Z)

Message 2

TO U BOAT ZERO TWO COOL DOWN AT TWO COMMA EIGHT

(move 7 code e.g. H = A, G = Z)

Message 3

TO U BOAT ZERO THREE CREW NEED TO SHOWER PROCEED TO FIVE COMMA SEVEN

(move 4 code e.g. E = A, D = Z)

U-Boat 01 goes to G – the row boat at (3,3)

U-Boat 02 goes to A – the iceberg at (2,8)

U-Boat 03 goes to D – the whale at (5,7)

The use of the word ‘code’ is not strictly correct. What most of us think of as a code is in fact a cipher. A code is a word or phrase with a prearranged meaning e.g. ‘the moon is full’ would signal that the army was ready for battle.

1. 1 minute

2. 16 coupons

3. Less (3 coupons)

4. 25 pieces

5. 4.55

6. One third

7. 3 pairs

8. 2 coupons

The letters read BLACKOUT

1. more than 9/10

2. 3/10

3. less than 1/10

4. 8/10

5. 5/10

6. 6/10

In order, the fractions give the secret word 'cloudy'.

Florence (1/2): 2/4, 7/14, 5/10, 3/6, 4/8, 6/12 Apple Farm

Percy (3/5): 6/10, 12/20, 15/25, 30/50, 18/30, 9/15 The Baker

Ethel (3/4): 6/8, 21/28, 15/20, 9/12, 12/16, 18/24 Post Office

Harry (2/3): 4/6, 12/18, 6/9, 8/12, 10/15, 14/21 The Butchers

1. S 2. P 3. U 4. D 5. B 6. A 7. S 8. H 9. E 10. R

which spells out SPUDBASHER.

2. one dozen (12) mugs of beef teas – R

3. 12 big pots of tea – O

4. 57 women (171 men) – I

5. (32 rhubarb) 80 gooseberry – C

6. 480 mugs of tea – T

7. One third of the day – V

The solution letters can be rearranged to spell ‘VICTORY’.

Jack has 20 marbles.

(Alan has 24, Ian has 22, Billy has 18 and Charlie has 16.

Plum pudding

The pudding weighs 500g.

The ingredients other than the honey sum to 95% of weight.

The honey therefore equals 5% of the pudding’s weight.

If 5% is 25g, then 100% is 500g.

Egg share

Mr Gibbons should visit Mrs Cummings first. This will ensure that for any number of eggs

over 20 he will retain 5 more eggs than if he visited the air-raid warden first. If he has less

than 20 eggs he must visit Mrs Cummings first or he will not be able to give her 10 eggs.

Sausage sell-out

Mr Bacon had 8 sausages at noon.

(4 + 1 to Mrs Bracknell and 3 sausages left for Mr and Mrs Bacon’s tea.)

Sid’s leave

Sid’s leave is for 48 hours.

Alice put out 12 rows of 20 chairs. That’s 240 seats.

Tickets cost £5.50 each (£22.00 ÷ 4). All three shows were sell outs,

so 720 tickets were sold bringing in £3960. TOP.

Only two creatures need to swop places, the snake with the bird.

Each side of the log will then be balanced at 12.5 kg.

Re-arranging four digits

If the zero had been 2, 4, 6, 7 or 9 there would have been a further six ways.

8530 5830 3850

8503 5803 3805

8053 5083 3085

8350 5380 3580

8305 5308 3508

8035 5038 3058

Multiplying the ‘gomutrika’ way

The correct products are: (C) 748 and (F) 58280.

The answers are: 1. (c), 2. (f), 3. (h), 4. (l), 5. (n), 6. (r), 7. (s), 8. (x), 9. (z).

Using the ‘Solutions Checker’ the word ‘bungalows’ is spelt out. Other Indian words

borrowed/adopted into English include: bangle, chintz, cot, dungaree, juggernaut, jungle,

loot, pyjamas, shampoo, thug.

To count the grains in just one sack, at the rate of one grain per second, it would take

Shashi 196 days (this assumes counting day and night and no toilet breaks for

more than six months).

From start (1 + 4) x 4 ÷ 2 + 40 - 1 = 49

From ruins (49 ÷ 7) x 6 ÷ 3 + 4 - 3 = 15

From pool (15 x 2) x 3 ÷ 9 + 50 - 10 = 50

From huts (50 - 5) x 20 ÷ 10 + 15 - 5 = 100 (Finish)

2. 16:40 &Mac221; B

3. false &Mac221; E

4. yes &Mac221; N

5. 8 km &Mac221; L

6. 3 km &Mac221; G

When rearranged the letters of the solutions spell out BANGLE.

This is one of many words borrowed from India (see solution to ‘A doubling or halving day’).

The solutions for this article need diagrams. Please email.

Changing money.

The answers are: No (G), Yes (A), No (N), Yes (G), No (E), Yes (S) and spells out GANGES.

The correct silhouette is (B).

Moving logs

The solutions for this article need diagrams. Please email.

Knotty snake

When stretched out the snake won’t have any knots.

Monkey puzzle

(A) belongs to Hari, (B) belongs to Mitwas, (C) belongs to Aish and (D) belongs to Jaz.

At ‘Indian Snacks’ the £3.00 will buy them seven samosas.

At ‘Spicy Eats’ the £3.00 will buy them only six samosas

with 20p left. TOP.