(Please note some solutions are dependent on images and are omitted here. If you need a
particular solution please e-mail).

Volume 7 (issues 19, 20 and 21)

CIRCA 19: Solutions

Front cover/The Pharaoh’s obelisk
1/2 + 1/3 + ? = 1, 3/6 + 2/6 + 1/6 = 1 1/6 = 5 cubits.

5 cubits x 6 = 30 cubits. The Pharaoh’s obelisk is 30 cubits tall.

You may interpret the question as requiring the height of the obelisk above the surface of the
water, in which case the answer is 15 cubits.

Pages 2 and 3/Finding fractions
Sharing apples
The two apples could be cut into halves (4 pieces), one of which is then divided into three pieces
(sixth of an apple). Each receives 1/2 + 1/6

Sharing chocolate
1/2 + 1/4 would seem to be the best division. Others are possible.
This problem can be extended to how best to share 3 chocolate bars between 5 people and so on.

Fraction parade
The characters with false ID’s are (b) and (d).
(b)’s numerator is greater than her denominator. This makes her statement untrue.
She is an improper fraction.
(d) is not improper. His numerator is neither the same nor greater than his denominator.
(d) is a common or vulgar fraction.

Pages 4 and 5/Egyptian fractions
Egyptian numbers
(a) 44500 (b) 1232143

Writing Egyptian fractions
(c) 1/537

Tohoser’s fraction problem
Tohoser’s initial idea for division of the canes would need 35 cuts
yielding 40 pieces. A better division needing less cuts is to halve four of the canes and divide
the remaining cane into eighths. This would need only 11 cuts with each child receiving
1/2 + 1/8.

Liquid fractions
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 63/64

Page 6 and 7/Unit fractions
(a) 1/7 = 1/8 + 1/56
(b) 1/10 = 1/11 + 1/110

(c) 1/4 = 1/5 + 1/20
(d) 1/20 = 1/21 + 1/420

Pages 8 and 9/Fraction Mansion Maze
Requires visual solution. Please e-mail.

Pages 10 and 11/Roll up! Roll up!
The big wheel that gives the ride of longest length is Tenpozan. The London Eye and the
proposed Voyager give the longest ride in time because Tenpozan stops for getting on and off.

Pages 12 and 13/Fnd the percentages
4 out of the sample of 10 are wearing roller skates.
4/10 = 40%. 40% of 2000 is 800 contestants

Pages 14 and 15/Are you puzzled?
Ernest’s equivalent fraction challenge
Guests 1, 2 and 3 all have half a cake in total on their plates already.
Ernest gives guest 4 the two quarters from plates 5 and 6 by swapping
the thirds with quarters.

Climbing pharaohs
A pharaoh takes 5 minutes to climb a tree. Seven pharaohs starting at the same
time would take 5 minutes each to climb a tree. A single pharaoh, assuming
he didn’t tire, would take 35 minutes to climb 7 trees one after the other.

Biggest total
The biggest total is 51.

Zasra’s mind reading trick
This trick involves manipulation of digital routes and some probability. No matter which number
(1 to 10) you choose, you get a multiple of 9. All digital routes of multiples of 9 are 9, so taking
away 5 will always leave 4, and hence, to D.

How many countries can you think of beginning with D? There are two, Denmark (the likely
choice) and Dominion Republic (less likely). Likely animals to come to mind beginning with E
are (in order of popularity of small sample) elephant (18), eel (0), emu (1). The Dominion
Republic was chosen by 1 person (out of 20) and the animal beginning with o was ostrich.
Other animals beginning with o are octopus, orang-utan, ostrich, otter, owl; there are others.

The life of Tut
Tut lived for 72 years. Start by adding all the fractions (baby = 1/36,
child = 1/6, school boy = 1/12, charioteer in the army = 1/18, wise pharaoh = 1/3).
They total 24—36 or 2–3. So, one third of Tut’s life is accounted for by the 24 years
he was a young pharaoh. 3 x 24 = 72.

Page 16 (back cover)/All change
Alice, assuming everyone turns up, will again partner Mira at the Monday session three weeks later. TOP.

CIRCA 20: Solutions

Front cover/Three bell hats
5 jesters are wearing hats with two bells and 6 jesters are wearing hats with three bells.

This puzzle can be solved by trial and improvement. There is an explanation of how to do it
and a further puzzle on page 15 (see solutions for ‘Are you puzzled?’).

For the sharp-eyed...
1. In the bottom left there is a copy of a magic square (arrangement of consecutive numbers,
usually starting at one, where all the rows, columns and both diagonals sum to the same total)
that can be found on Du¨rer’s engraving of 1514, Melancolia. Some of the numbers are obstructed.
Which ones? What is the total? For further information on magic squares see CIRCA 11.

2. The banners hanging from the balcony provide a puzzle/sorting activity involving shape
on page 11 (see solution The family tree of the Quadrilaterals).

3. There are also various characters to spot who can also be found within the magazine,
e.g. Shelley from page 8, the ‘mathmagician’ from page 6, etc.

Pages 2 and 3/Making turns
The spoon is on the left of the boy and so (2) is the correct drawing.

The jesters do a turn...
All but one of the jesters is holding the wrong card. This is how it should be:
Greg Grinning should be holding card D,
Joking Jane should be holding card B,
Laughing Laura is holding the correct card (C),
Stan Smile should be holding card E, and
Will Wisecracker should be holding card A.

Pages 4 and 5/The missing angle

A = 78° (90° + 12° + 78° = 180°)
B = 35° (90° + 55° + 35° = 180°)
C = 60° (90° + 30° + 60° = 180°)

Page 6 and 7/Maths magic
An explanation of how the trick works is given on page 7.
A similar trick appeared in CIRCA 14 using dominoes.
It is an interesting exercise to work out how the two tricks are connected.
“Choose any domino from a double six set (6/6, 6/5, 6/4 ... 1/3, 1/2, 1/1, 1/0, 0/0).
Don’t tell me which!”
“Pick one of the numbers on the domino. Multiply it by 5”.
“Add 7 and then double the total.”
“Add the number from the other half of the domino.”
“Subtract 14”
“You are thinking of a 2-digit number. The first digit is the number of spots on one half of
the domino you chose. The other digit is the number of spots on the other half.”

Pages 8, 9 and 10/Cool pool
Please e-mail for solution.

Page 11/The family tree of the Quadrilaterals
The banner of Sir Thinkalot is a blue trapezium.

Pages 12 and 13/Angle Castle Maze
Please e-mail for solution.

Pages 14 and 15/Are you puzzled?
Which silhouette?
(C) is the correct silhouette.

Starboard and port
Needs visual solution. Please e-mail.

Which door?
Door 2 has the biggest number (582). Door 1 was 155 and door 2 was 509.5.

Regular, light or Whatever
Only £1.00 is needed to be sure of getting a light cola, the next time you put a coin in.

Here’s why:
Because the buttons never give what their labels say, if you press the ‘Whatever’ button
and get a regular, you will know that this button will always provide regular cola. Pressing
the ‘Light’ button will therefore give you random colas (i.e. ‘Whatever’) and pressing ‘Regular’
will give you light. Alternatively, if the ‘Whatever’ button gives you a light, you can deduce
that this button will always provide light colas.

Trial and improvement
A knight at the theatre
The 77 pennies bought tickets for 13 adults and 4 children.

Page 16 (back cover)/Spending euros
Alice postcard 1.50 euro
Blake postcard/keyring 3.48 euro
Courtney pen/snowscene 6.90 euro
Denise keyring/pen 4.98 euro

From the information in the cartoon we can find out which souvenirs Alice, Blake and Courtney
bought. By taking the amount of the change from the E20 note that Alice paid with we know how
much the group spent in total. Subtracting the spending of the other three gives us the amount
spent by Denise. Only two of the items on the souvenir stall sum to that amount. TOP.

CIRCA 21: Solutions

Front cover/How fast?
After one second engine 1 on its own would have propelled the spaceship 1/20 of the distance
to Planet Zargo. In the same time engine 2 could have propelled the craft 1/30 of the distance.
Used together they would travel 1/20 + 1/30 of the distance in a second. Therefore the ship
would travel 1/12
( 1/20 + 1/30 = 5/60 = 1/12 ) the distance in a second.
So, it would take the aliens 12 seconds to reach Planet Zargo.

Pages 2 and 3/Finding ratios in Uma’s pencil case
1. (c) 3:2 2. (a) 5:2 3. (c) 2:3 4. (a) 1:1 5. (c) 1:2

Ratio reckoning
(1) C (2) D (3) A

(4) E (5) F (6) B

Pages 4 and 5/The Crumbles get painting

The correct ratio of yellow paint to blue is 2:1.
Mixing half a litre of blue with the remaining litre of yellow would give the correct mix.
Half a litre of blue paint will be left.

Pages 6 and 7/How tall was Mona Lisa?
Getting Vitruvian Man in proportion
Measuring the reproduced and reduced version can be slightly inaccurate. Pupils should find
(1) height 91.5mm and span 92mm making both almost equal.
(2) Head 13mm. The head divides into the height approximately 7 times.
(3) Since Leonardo’s time artists have generally accepted an average ratio of about 7:1 of
head lengths to height of grown men and women.

Lisa’s height
The length of Lisa’s head is 208mm. 208mm x 7 = 1456mm –
making Lisa 1.456 metres tall (4ft 9ins).

Changing proportions
Ratios that children find if they measure each other will vary from Vitruvian Man as the proportions
of a child change throughout childhood. A new born child will have short limbs and a relatively large
headsize. As the child gets older head growth slows (brain size growth more-or-less stops at
age 5) but limb growth speeds up.

Pages 8 and 9/Star path multiples maze
Purple (4) goes to Zumbos.
Blue (7) goes to Zom.
Green (3) goes to Ziggus.
Yellow (6) goes to Zargo.

Pages 10 and 11/Digit-daisy chains
Longest and shortest chains
Count all the daisies in the chain including the starting pair but excluding the finishing repeated pair.
(0, 0) will be the shortest = 2 daisies
(0, 5) = 3 daisies
(2, 6) = 4 daisies
(0, 1) = 60 daisies
as does (1,0). 60 is the longest chain.

Starting pairs
There are 100 possible starting pairs.
First daisy has 10 possibilities 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0.
Each of these has 10 possibilities e.g. (1,1), (1,2), (1, 3) and so on. Stress to the children that
they have to count ordered pairs because swapping around the numbers could give different results
[e.g. (2,1) produce a shorter chain than (1,2).] except in the case of repeated digits [e.g. (2,2)].

Pages 12 and 13/Billy’s bathtime
The check letters should read A CLEAN BILLY

Other stories... other graphs
Graph Title y-axis
A 2 (iii)
B 1 (ii)
C 3 (i)

Pages 14 and 15/Are you puzzled?
Which cube?
(d) is the cube made when the net is folded-up.

Which card is which?
(A) is the Jack of Clubs
(B) is the Queen of Hearts
(C) is the four of diamonds

Cistern problems
Tara and Brenda could consume the cola in 2 minutes.
(In one minute they will drink 1/3 + 1/6)

Pages 16 (back cover)/Rosie’s birthday
There will be 10 ‘clinks’ for the party of five (each member ‘clinking’ once only with
every other member).

Alice clinks with Louis, Gina, Howie and Rosie A= 4
Louis clinks with Gina, Howie and Rosie L= 3
Gina clinks with Howie and Rosie G= 2
Howie clinks with Rosie H= 1
Rosie doesn’t initiate any ‘clinks’ R = 0