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

particular solution please e-mail).

**Volume ****8 (issues 22, 23 and 24)**

**CIRCA 22: Solutions**

**Front cover/Sam on the run**

121 bricks were knocked out by Sam.

**Pages 2 and 3/Sam sums some eggs**

12 hens

12 hens producing in the way described would lay 78 eggs.

See page 15 for more on summing series of integers beginning with 1.

Does the method work for an odd number of terms such as

1 + 2 + 3 + 4 + 5 + 6 + 7? Yes! 3 1/2 x 8 = 28. Why? See answers for page 15.

**Pages 4 and 5/Mobile maths**

All the phone callers have a partner. They are:

A is talking to K F is talking to R

B is talking to P G is talking to L

C is talking to J H is talking to T

D is talking to N I is talking to O

E is talking to M S is calling to Q

**Page 6 and 7/Good guessing**

Making a guess is often the best approach for starting to solve a problem.

Right or wrong, it gives a point of reference from which to refine the guess.

Checking that the answer you give fits the facts allows for revision of answers

and assurance that an answer is correct.

Tubes of paint

Renée used 14 tubes of blue, 25 tubes of red and 9 tubes of yellow.

Renée’s brushes

Renée has 9 hogs-hair brushes,

4 sable brushes and 12 nylon brushes.

Getting the right balance

(a) 60 + 17 = 77 and 55 + 22 = 77 Rogue number: 23

(b) 92 + 75 = 167 and 94 + 73 = 167 Rogue number: 84

(c) 35 + 65 = 100 and 77 + 23 = 100 Rogue number: 79

(d) 24 + 59 = 83 and 44 + 39 = 83

(24 + 59) + (44 + 39) = 166 and 83 + 83 = 166

Rogue number: 92.

**Pages 8 and 9****/Nim Tec**

Nim Tec is a variation of ‘Nim’ (meaning ‘take’ in German). Thought to have originated in China,

it was first recorded in Europe in the 15th Century, and its theory was analysed by Bouton of Harvard

University about 100 years ago. Many mathematicians have since created ‘Nim’ type games.

The game of Nim is probably more correctly described as a puzzle rather than a game. By playing

a ‘best game’ the players can say who will win at the start of play. Proofs of this can be shown with

Boolean arithmetic and binary representation, but here we demonstrate how Player 2 can win at

Nim Tec if we just follow play on one wheel:

Player 1 moves to 1, 2 or 3 Player 2 moves to 4

Player 1 moves to 5, 6 or 7 Player 2 moves to 8

Player 1 moves to 9, 10 or 11 Player 2 moves to 12

Player 1 moves to 13, 14 or 15 Player 2 moves to 16

Player 1 moves to 17, 18 or 19 Player 2 moves to 20

With two or more wheels, Player 2 can repeat her strategy, as she will always have the option of

moving 1, 2 or 3 to be sure of landing the counters on multiples of 4 on her turns.

**Page 10 and 11****/Times for the table**

Mental arithmetic techniques to calculate the times-tables can be a great help to pupils who

‘panic’ with certain numbers. The seven times-table is used here to illustrate the use of doubling,

halving and subtraction – these can be used for all multiplication tables.

**Pages 12 and 13****/Intruder**

The intruders spacecraft is number 37. All the other numbers can be made using multiples

of 3 and 20. For example:

50 = 1 x 20 + 10 x 3

26 = 1 x 20 + 2 x 3

65 = 1 x 20 + 15 x 3

167 = 7 x 20 + 9 x 3

253 = 11 x 20 + 11 x 3

47 = 1 x 20 + 9 x 3

32 = 1 x 20 + 4 x 3

79 = 2 x 20 + 13 x 3

181 = 8 x 20 + 7 x 3

67 = 2 x 20 + 3 x 3

1 to 100

Between 1 and 19 only the multiples of 3 can be made, 3, 6, 9, 12, 15 and 18.

All the numbers between 20 and 100 can be made except 22, 25, 31, 34 and 37.

Over 100

All numbers over 100 can be made with multiples of 3 and 20.

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

Which is longer?

Both lengths are the same.

Next pile

The piles of bricks show the beginning of the Fibonacci Numbers. Each new term is the

sum of the previous two terms. The next number in the series is 13.

Chicken bricks

The chasing chicken knocked out 68 bricks.:

Where will it land?

Needs visual solution. Please e-mail.

Broken key

There are many ways to solve Shreena’s problem. One way is divide by 68 then double.

4862 ÷ 68 = 71.5 71.5 x 2 = 143

Wolfman

The pictures were taken in the order 5 (normal), 2 (ears growing), 6 (ears bigger and hands

changing to claws), 1 (hands more claw like and snout beginning), 4 (snout bigger and teeth

appear) and finally 3 (teeth and head bigger, clothes tearing).

1 to 1000000

The sum of 1 to 100 is 5050.

The sum of 1 to 110 is 6105.

The sum of 1 to a billion (US billion of 1000 millions) is

500 000 000 500 000 000

Summing series ending in odd numbers

If the two methods shown here are used then a multiplication of a fraction is needed (e.g. for the

series 1-7, the half-way number is 3.5). However, the sum can be more easily calculated by

multiplying the last number by the first whole number after the midpoint (e.g. 4) as this represents

the number of pairs that total to the last number (e.g. 4 x 7 = 28).

**Page 16 (back cover)****/How far?**

Good guessing

To solve this problem it’s important to bear in mind that the journey took place in an afternoon,

which limits the length of journey that could be made. A good start would be to question how far

could a coach travel in about an hour: 50 miles? 60 miles?

Trial and improvement

Using the guess, trial and improvement method, the problem could be solved:

Trial 1 Trial 2 Trial 3

21912 21912 21912

+ 100 + 90 + 110

22012 22002 22022 ¨ palindrome

The school is 55 miles from the site of the visit.

Alternatively, a search for the next palindromes would bring up 22022 and 22122, and taking

into account the likely mileage, it would suggest the first to be the correct answer.

TOP.

**CIRCA 23: Solutions**

**Front cover/Odd one out**

The odd bull is a reflection of all the others, which are rotations of each other. The odd bull

is the bull, with legs more-or-less pointing upwards, directly north-east of the bottom bull.

**Pages 2 and 3/Divide the lot**

A (3) got the modern painting (£231).

N (5) got the vase (£355).

T (9) got the antique mirror (£1746).

I (100) got the model plane (£2500).

Q (8) got the knight’s helmet (£3152).

U (2) got the stuffed fish (£14).

E (6) got the antique globe (£222).

S (4) got the bagpipes (£172).

The bidders letters spell out ANTIQUES.

**Pages 4 and 5/****Prime time**

Prime numbers under 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,

67, 71, 73, 79, 83, 89, 97

Prime numbers 100 – 200

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181,

191, 193, 197, 199

Prime patterns

Prime numbers greater than 3 will always fall in the columns headed 1 and 5 in a 6-column grid.

All primes greater than 3 are 1 less or 1 more than a multiple of 6.

Twin primes

Pairs of prime numbers that differ by 2 are called ‘twin primes’. 11, 13 and 137, 139 are examples.

Though the occurrence of twin primes becomes rarer as numbers get larger there is an infinite

number of them. The largest twin primes discovered so far (in 2002) written out would contain

over 50000 digits each.

Digitally interesting

11 is the lowest prime number made of the same digit. 1111111111111111111 (19 ones)

and 1111111111111111111111(23 ones) are both prime.

**Pages 6 and 7/Can aliens read your mind**

The alien’s trick involves making a number that sums to the current year plus a bit more:

800 + 1603 = 2503

When you subtract your birth year you will be left with

your current age and your original number, x. The process of doubling and adding 8,

then multiplying the total by 50 is effectively the same as multiplying the original

number, x, by 100.

So, why does the alien have x? Because if she just asked you to add 400 (8 x 50) to 1603 it

would be easy to see what was going on. By using x, which can vary, there are different

sums at the end and 2003 is not obvious.

By doubling and multiplying by 50 (making x times 100) the single digit of x is moved into

the hundreds column.

**Pages 8 and 9****/Add ‘em up**

Sum hoops

Three ways to score 18:

(A) 6 + 6 + 6 (B) 2 + 8 + 8 (C) 2 + 7 + 9

Three ways to score 20:

(A) 6 + 6 + 8 (B) 2 + 9 + 9 (C) 6 + 7 + 7

Sum darts

Three ways to score 200:

(A) 100 + 40 + 40 + 20 (B) 50 + 50 + 50 + 50

(C) 100 + 50 + 40 + 10

Three ways to score 150:

(A) 50 + 50 + 40 + 10 (B) 100 + 20 + 20 + 10

(C) 50 + 40 + 40 + 20

Sum buckets

Three ways to score 600:

(A) 200 + 200 + 125 + 75 (B) 300 + 150 + 75 + 75

(C) 200 + 150 + 125 + 125

Two ways to score 450:

(A) 125 + 125 + 125 + 75 (B) 150 + 150 + 75 + 75

**Pages 10 and 11****/Counting rods**

A = 3, B = 1, C = 4, D = 2 and E = 5.

particular solution please e-mail).

121 bricks were knocked out by Sam.

12 hens producing in the way described would lay 78 eggs.

See page 15 for more on summing series of integers beginning with 1.

Does the method work for an odd number of terms such as

1 + 2 + 3 + 4 + 5 + 6 + 7? Yes! 3 1/2 x 8 = 28. Why? See answers for page 15.

All the phone callers have a partner. They are:

A is talking to K F is talking to R

B is talking to P G is talking to L

C is talking to J H is talking to T

D is talking to N I is talking to O

E is talking to M S is calling to Q

Making a guess is often the best approach for starting to solve a problem.

Right or wrong, it gives a point of reference from which to refine the guess.

Checking that the answer you give fits the facts allows for revision of answers

and assurance that an answer is correct.

Tubes of paint

Renée used 14 tubes of blue, 25 tubes of red and 9 tubes of yellow.

Renée’s brushes

Renée has 9 hogs-hair brushes,

4 sable brushes and 12 nylon brushes.

Getting the right balance

(a) 60 + 17 = 77 and 55 + 22 = 77 Rogue number: 23

(b) 92 + 75 = 167 and 94 + 73 = 167 Rogue number: 84

(c) 35 + 65 = 100 and 77 + 23 = 100 Rogue number: 79

(d) 24 + 59 = 83 and 44 + 39 = 83

(24 + 59) + (44 + 39) = 166 and 83 + 83 = 166

Rogue number: 92.

it was first recorded in Europe in the 15th Century, and its theory was analysed by Bouton of Harvard

University about 100 years ago. Many mathematicians have since created ‘Nim’ type games.

The game of Nim is probably more correctly described as a puzzle rather than a game. By playing

a ‘best game’ the players can say who will win at the start of play. Proofs of this can be shown with

Boolean arithmetic and binary representation, but here we demonstrate how Player 2 can win at

Nim Tec if we just follow play on one wheel:

Player 1 moves to 1, 2 or 3 Player 2 moves to 4

Player 1 moves to 5, 6 or 7 Player 2 moves to 8

Player 1 moves to 9, 10 or 11 Player 2 moves to 12

Player 1 moves to 13, 14 or 15 Player 2 moves to 16

Player 1 moves to 17, 18 or 19 Player 2 moves to 20

With two or more wheels, Player 2 can repeat her strategy, as she will always have the option of

moving 1, 2 or 3 to be sure of landing the counters on multiples of 4 on her turns.

‘panic’ with certain numbers. The seven times-table is used here to illustrate the use of doubling,

halving and subtraction – these can be used for all multiplication tables.

of 3 and 20. For example:

50 = 1 x 20 + 10 x 3

26 = 1 x 20 + 2 x 3

65 = 1 x 20 + 15 x 3

167 = 7 x 20 + 9 x 3

253 = 11 x 20 + 11 x 3

47 = 1 x 20 + 9 x 3

32 = 1 x 20 + 4 x 3

79 = 2 x 20 + 13 x 3

181 = 8 x 20 + 7 x 3

67 = 2 x 20 + 3 x 3

1 to 100

Between 1 and 19 only the multiples of 3 can be made, 3, 6, 9, 12, 15 and 18.

All the numbers between 20 and 100 can be made except 22, 25, 31, 34 and 37.

Over 100

All numbers over 100 can be made with multiples of 3 and 20.

Both lengths are the same.

Next pile

The piles of bricks show the beginning of the Fibonacci Numbers. Each new term is the

sum of the previous two terms. The next number in the series is 13.

Chicken bricks

The chasing chicken knocked out 68 bricks.:

Where will it land?

Needs visual solution. Please e-mail.

Broken key

There are many ways to solve Shreena’s problem. One way is divide by 68 then double.

4862 ÷ 68 = 71.5 71.5 x 2 = 143

Wolfman

The pictures were taken in the order 5 (normal), 2 (ears growing), 6 (ears bigger and hands

changing to claws), 1 (hands more claw like and snout beginning), 4 (snout bigger and teeth

appear) and finally 3 (teeth and head bigger, clothes tearing).

1 to 1000000

The sum of 1 to 100 is 5050.

The sum of 1 to 110 is 6105.

The sum of 1 to a billion (US billion of 1000 millions) is

500 000 000 500 000 000

Summing series ending in odd numbers

If the two methods shown here are used then a multiplication of a fraction is needed (e.g. for the

series 1-7, the half-way number is 3.5). However, the sum can be more easily calculated by

multiplying the last number by the first whole number after the midpoint (e.g. 4) as this represents

the number of pairs that total to the last number (e.g. 4 x 7 = 28).

To solve this problem it’s important to bear in mind that the journey took place in an afternoon,

which limits the length of journey that could be made. A good start would be to question how far

could a coach travel in about an hour: 50 miles? 60 miles?

Trial and improvement

Using the guess, trial and improvement method, the problem could be solved:

Trial 1 Trial 2 Trial 3

21912 21912 21912

+ 100 + 90 + 110

22012 22002 22022 ¨ palindrome

The school is 55 miles from the site of the visit.

Alternatively, a search for the next palindromes would bring up 22022 and 22122, and taking

into account the likely mileage, it would suggest the first to be the correct answer.

TOP.

The odd bull is a reflection of all the others, which are rotations of each other. The odd bull

is the bull, with legs more-or-less pointing upwards, directly north-east of the bottom bull.

N (5) got the vase (£355).

T (9) got the antique mirror (£1746).

I (100) got the model plane (£2500).

Q (8) got the knight’s helmet (£3152).

U (2) got the stuffed fish (£14).

E (6) got the antique globe (£222).

S (4) got the bagpipes (£172).

The bidders letters spell out ANTIQUES.

Prime numbers under 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,

67, 71, 73, 79, 83, 89, 97

Prime numbers 100 – 200

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181,

191, 193, 197, 199

Prime patterns

Prime numbers greater than 3 will always fall in the columns headed 1 and 5 in a 6-column grid.

All primes greater than 3 are 1 less or 1 more than a multiple of 6.

Twin primes

Pairs of prime numbers that differ by 2 are called ‘twin primes’. 11, 13 and 137, 139 are examples.

Though the occurrence of twin primes becomes rarer as numbers get larger there is an infinite

number of them. The largest twin primes discovered so far (in 2002) written out would contain

over 50000 digits each.

Digitally interesting

11 is the lowest prime number made of the same digit. 1111111111111111111 (19 ones)

and 1111111111111111111111(23 ones) are both prime.

The alien’s trick involves making a number that sums to the current year plus a bit more:

800 + 1603 = 2503

When you subtract your birth year you will be left with

your current age and your original number, x. The process of doubling and adding 8,

then multiplying the total by 50 is effectively the same as multiplying the original

number, x, by 100.

So, why does the alien have x? Because if she just asked you to add 400 (8 x 50) to 1603 it

would be easy to see what was going on. By using x, which can vary, there are different

sums at the end and 2003 is not obvious.

By doubling and multiplying by 50 (making x times 100) the single digit of x is moved into

the hundreds column.

Sum hoops

Three ways to score 18:

(A) 6 + 6 + 6 (B) 2 + 8 + 8 (C) 2 + 7 + 9

Three ways to score 20:

(A) 6 + 6 + 8 (B) 2 + 9 + 9 (C) 6 + 7 + 7

Sum darts

Three ways to score 200:

(A) 100 + 40 + 40 + 20 (B) 50 + 50 + 50 + 50

(C) 100 + 50 + 40 + 10

Three ways to score 150:

(A) 50 + 50 + 40 + 10 (B) 100 + 20 + 20 + 10

(C) 50 + 40 + 40 + 20

Sum buckets

Three ways to score 600:

(A) 200 + 200 + 125 + 75 (B) 300 + 150 + 75 + 75

(C) 200 + 150 + 125 + 125

Two ways to score 450:

(A) 125 + 125 + 125 + 75 (B) 150 + 150 + 75 + 75

**Pages 12 and 13****/Ice screamer
**Paula’s position in the queue reduces by two each time someone is served, the person served

plus the person pushed by. Children should therefore discover that the number served before

Paula makes it to the front is 15. For all such queues with the same conditions (i.e. an even

number in the queue) the result will be half the queue. Where the queue is an odd number of

people then it’s the highest multiple of 2, e.g. 31 in queue and 15 people served before Paula

gets to front.

1. A 2. B 3. A 4. B

Divide the lot!

1 (2–3 of 3–4 = 1–2 + 1–2 = 1).

Find the spy

The spy is, starting from the top left: row 4, column 6.

How many?

12 chocolates..

Which is next?

Each symbol is made by using a numeral and its reflection.

The line of numbers is 1, 2, 3, and 4; the next in line is 5, the first of the designs.

Twice as much

The one-eyed alien must give 10 dollars to the three-eyed alien.

The one-eyed alien will then have 20 dollars and the three-eyed alien will have 40 dollars.

1 = 2, 2 = 5, 3 = 5, 4 = 4, 5 = 5, 6 = 6, 7 = 3, 8 = 7, 9 = 6, 0 = 6

Poona brought the food over to Alice at 17:48 (12 minutes to 6 pm).

After 30 minutes the time was 18:18 (18 minutes past 6 pm).

The time of day with most lines 08:08 (8 minutes past 8 am). It uses 28 lines.

The time of day using the least number of lines is 11:11

(11 minutes past 11 am). It uses only 8 lines.

TOP.

The mirror image is number 6. (1. strap missing on front leg;

2. belt missing; 3. sandal leathers reduced; 4. head-band missing;

5. money bag missing; 7. bracelet missing; 8. leg straps missing;

9. finger not pointing.)

Crossways (26 km) to Windy Hill (16 km) back to the Swan Inn (8 km).

Route 2 (takes 11 hours at 10 km per hour; 110 km): Swan Inn to Marsh Point (40 km) to

Lighthouse Point (6 km) to Ports Town (14 km) to Castle Town (24 km) back to

the Swan Inn (26 km).

Route 3 (takes 12 hours at 10 km per hour; 120 km): Swan Inn to Marsh Point (40 km)

to Lighthouse Point (6 km) to Ports Town (14 km) to Castle Town (24 km) to

Crossways (12 km) to Windy Hill (16 km) back to the Swan Inn (8 km).

(B) Bad Bob never gets to rob the mail coach. None of the routes can go along the road between

Castle Town and Marsh Point in this puzzle as the route would result in an odd number of

kilometres travelled and all the routes have an even total.

There could be some arguments about which measure matches which item but, by elimination,

children should arrive at the following ‘agora shopping list’:

rope M fathoms

cloth A cubits

belt R feet

cabbage K span

figure E palms

nails T digits

The mystery word is ‘MARKET’.

The drawing of the ‘Measuring man’ is based on the ‘Metrological Relief’ that can be found at

the Ashmolean Museum in Oxford. The table shows the more common measurements of

ancient Greece. The ‘Metrological Relief’ shows measurements that are local to the part of

Greece in which it was carved and are slightly different.

cm equivalents and a full-size photograph of a man’s hand to measure.

The discovery that the width of one’s own thumb is the same as its length leads on to knowing

the length of the statue’s thumb’s length because we know it was the equivalent of a fathom

(a man could just reach around it). The table on page 5 tells us that a fathom was 177.6 cms.

The photograph on page 5 can be used to measure how many times a digit (the width of the

middle finger) goes into a thumb,either with a ruler or a marked piece of paper.

Having agreed with an estimate of 4 times, the width of Colossus’ finger can be worked out by

dividing 177.6 by 4.

This gives us Colossus’ equivalent of a Greek digit as 44.4 cms.

Using the information in the table on page 5:

One Greek foot equals 14 digits,

so, Colossus’ foot is 44.4 x 14 = 621.6 cms

One Greek fathom equals 6 feet,

so, Colossus’ equivalent of a fathom is 621.6 x 6 = 3729.6 cms

Both the cartoon on page 7 and the illustration by Leonardo on the worksheet illustrates that

a fathom is the same length as a man’s height, giving us the height of 37.30 metres

(122 ft) for the statue of Colossus. (This does not include the height of the plinth.)

Alternatively, pupils can use ratios of their thumb length to their height to compare

Colossus’ thumb and height.

A nice extension to this activity is to get the child to work out how tall a statue of themselves

would be if it was built to the same scale as Colossus.

See also ratio and estimating Mona Lisa’s height in CIRCA, Issue 21, Volume 7.

Achilles started with 6 coins in his pocket.

At finish of first run 6 x 2 = 12

After paying old man 12 – 8 = 4

At finish of second run 4 x 2 = 8

After paying old man 8 – 8 = 0

(In case you didn’t notice, the secret words are an old saying ‘A fool and his money are

soon parted’ spelt backwards.).

A 5 180° 18.00

B 6 45° 19.30

C 4 120° 8.00

D 1 60° 22.00

E 3 30° 13.00

F 2 90° 15.00 .

gum 1 9 9

gravy 2 9 18

ice cream 5 9 45

chocolate 3 9 27

custard 6 9 54

Total 21 189

Tara’s nan has enough ‘Stain Buster’; there will be 11ml left in the bottle.

The missing number is 381. In each row the middle number (median) is halfway between

those on either side.

Bad Bob’s missing bit

The missing bit of hat is 3.

Pencil place

(I) is the longest and (G) is the shortest.

In order, longest first,

(I) 160mm,

(F) 147mm,

(A) 142mm,

(E) 130mm,

(D) 124mm,

(C) 120mm,

(B) 115mm,

(J) 111mm,

(H) 105mm,

(G) 100mm.

Little or large

It’s relative!

Around the palace

Please e-mail for a solution.

End-to-end

Please e-mail for a solution.