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

Volume 5 (issues 13, 14 and 15)

CIRCA 13: Solutions

Front cover/Starship maze and Cosmic maze:

Pages 2 and 3/Numbers on things
Digit detective: The two wrong bar code numbers are on the chocolate bar and the beans.
Both contain a transposition error.

Chocolate should be 5 023247 047329 and
Beans should be 5 000157 024886.

Pages 4 and 5/Remainder mine
Hidden message: YOU HAVE FOUND THE GOLD

Page 6 and 7/Where on the number line?
There are no answers required but did you know the first 10-digit sequence of all different
digits in &Mac185; is 2109876543? The zero in this sequence is the 26160636 th decimal place of &Mac185;.

Pages 8 and 9/Cross the swamp
9 to 63 to 45 to 72 to 99 to 27 to 81 to 54 to 9
7 to 63 to 28 to 49 to 84 to 70 to 7
5 to 30 to 45 to 25 to 40 to 70 to 5
8 to 24 to 80 to 72 to 40 to 56 to 8

Pages 10 and 11/Times and times again...
The diagonal on the ‘Pythagorean Table’ shows square numbers.

Multiplication Table Puzzles:
By multiplying smaller numbers then adding.
8 x 8 = 64, 64 + 64 = 128

18 appears four times on the 10 x 10 multiplication square.
Its factors are 2, 3, 6 and 9.

By inversing the operation. Find 56 on the table and find out if 7 is a factor.
If it is, then the other factor provides the answer.

The diagonal from top left to bottom right as (Pythagorean Table).

No prime numbers over 10 because they can’t be made by other numbers except themselves
and one. It would have to be a 97 x 97 square because this is the largest prime number
under 100.

11 and 12 times tables were learnt because of the old British system. eg 12 pennies in a
shilling and 12 inches to a foot.

Pages 12 and 13/Russian multiplication
A is wrong

Pages 14 and 15/Are you puzzled?
Sixth Avenue
A, B, E and F are possible (there are several routes for each one.
C and D cannot be done.

Spiral to square

Aging problem
Shanta is 7 years old.
7 + 5 = 12
7 - 3 = 4
4 x 3 = 12

The soda palace
Alf puts two waffles on the grill.
He grills one side of waffle 1 and waffle 2 (time taken 2 mins).

Alf turns over waffle 1, removes waffle 2 and replaces it with waffle 3.
He grills the second side of waffle 1 and one side of waffle 3 (time taken 2 mins).

Alf removes waffle 1 and replaces it with waffle 2 (side ungrilled), turns over waffle 3.
He grills the second side of waffle 2 and second side of waffle 3 (time taken 2 mins).

Total grilling time for three waffles is six minutes.

Maud's doughnuts
Move left and right doughnuts from top row down to bottom row.

Which bag?
Of the three bags one contains 44 buttons, one bag contains 41 buttons and the third 25 buttons.
If we say bag 1 contains 44, bag 2, 41 and bag 3, 25 the following are correct:
BAG 1: A, E, G, I
BAG 2: B, C, F, L
BAG 3: D, H, J, K, M, N
The bags and amounts of buttons need not be in the order given here.

Page 16 (back cover)/Book prize
The five different ways of giving the books are

4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1

From the fifth frame, we know that Alice received 4 books. From the fourth frame, Bella
receives 2 books and so does Carlos, so one of them must get (2 + 2) and the other must
get (2 + 1 + 1).

From the fifth frame, we also know that Ellie receives 3 books, so she must get the (3 + 1)
combination. Darren must receive the (1 + 1 + 1 + 1) combination (which is one book from
each of the others). TOP.

CIRCA 14: Solutions

Front cover/Choose a dominoThe explanation for Zasra’s cover trick is explained on page 2
of the magazine. (This will work for any pair of single-digit numbers.).

The explanation for Zasra’s second trick is given on page 14 using examples of odd and even
numbers. For an algebraic explanation of why this works see below (answers for page 14).

Pages 4 and 5/
Which is best?
The posters were displayed:
Week 1 = C
Week 2 = A
Week 3 = B
But it could be argued that week 1/week 2 were A/C
respectively.

Fraction A, C and E
Decimal D, H and I
Percentages B,F and G

Page 6 and 7/Getting 100%
80%. (The image needs to be one fifth smaller.)

Pages 8 and 9/Mirror maze
Starting from top left, 4th turtle along, D(own), R(ight), R, D, D, R, D, D and out.

Page 10 and 11/Mansion and matrix
Sir Percy is in chains, Eliza is silent and Ol’ Henry is headless.

The furry monster of Darkley Woods is Ark, Boggle is the wobbly one and Claw is slimy.
The crucial link is to recognise that Boggle likes the slimy monster but doesn’t like Ark.

Pages 12 and 13/New Town taxi cab distances
Passenger
Assuming both cars are the same and both drivers travel at the same speed then the car
nearest the station should be sent. Car Alpha is 9 blocks away and Car Bravo is 10 blocks away

Moving
The length of journey for Ben and Ling can be unequal. The shortest distance between the
hospital and the school is 12 blocks, so anywhere that is no further is OK. When they move,
a flat at (4,5) or (5,4) would be ideal although anywhere along the diagonal would satisfy the
condition of equal distances.

Taxi parking
(5,7) is the best because the total of the three distances from here is 12.

Pages 14 and 15/Are you puzzled?
Eight eights
888 + 88 + 8 + 8 + 8 = 1000

A further sum
12111

Odd one out
It is possible to make each of the number signs an odd one out:
The first sign is odd because it is the only one with 6.
Bottom left is odd because it is in a triangle.
Bottom right is odd because the number is white on black.
Top left is odd because there is nothing odd about it.

Clock logic
The following reflections are the same as the real time on the digital clock:

00:00 10:01 20:05
01:10 11:11 21:15
02:50 12:51 22:55
05:20 15:21

Soccer logic
No, Judy did not score the goal.
If Sally is truthful, Judy didn’t score . If Judy tells the truth, Sally can’t also be telling the
truth – so Judy is lying.

If Kathy is lying, they are both truthful (impossible!) or they are both lying (also impossible!).
So Kathy is telling the truth and Kitty must be lying.

Odd or even (page 14)

In the following explanation n stands for any starting number.

n ...we’ll choose 15

3n ...that makes 45

If n is even, 3n is also even. If n is odd, 3n is also odd.
If it is even, Zasra says divide by 2...

11–2n

(If it’s odd, Zasra says add 1 and then divide by 2...

11–2n + 1–2) ...our example gives 46 ÷ 2 = 23

Treble the result...

41–2n (or 41–2n + 11–2) ...that makes 69

...and divide by 9...

1–2n (or 1–2n + 1–6) ...that makes 7, remainder 6

and ignore any remainder

1–2n ...the result, 7

Because Zasra asks if the starting number was odd or even, she knows whether to add 1
after doubling the result. Hence, she can say the number first thought of.

Can you match the fractions (page 15)

A, D, F and K are the same. B, E, H and I are the same.
C, G, J and L are the same.

Page 16 (back cover)/Which way?
Alice knew they had come from Crofton Down so she set the signpost so that it pointed to
were they had come from. The other directions were then correct. TOP

CIRCA 15: Solutions

Front cover/Space Hops
The least number of moves needed is 15.

Pages 2 and 3/Sorting numbers
64 Evenly-even
52 Oddly-even
35 Oddly-odd
12 Oddly-even
98 Oddly-even
17 Prime Number
128 Evenly-even
50 Evenly-odd

(Oddly-even numbers are divisible by 4 but Evenly-odd
numbers aren’t.)

Prime numbers
2 (first and only even prime number), 23 and 15

Pages 4 and 5/
The perfect crime
The two perfect numbers under 30 are 6 (1 + 2 + 3) and 28 (1 + 2 + 4 + 7 + 14).

Page 6 and 7/Crazy factor golf
(a) Windmill = 9 (b) Castle = 18 (c) Lighthouse = 3
(d) Tunnel = 7 Spaceship = 6
No other combination is possible.

Pages 8 and 9/Get to the cheese

Pages 10 and 11/Nim
Nim is in the class of games like tic-tac-toe, noughts and crosses and draughts where strategy
is the only element that counts. In the game of Nim shown (there are numerous variations and
playing pieces vary from stones to shells) one of the players can always win provided that she
always plays ‘best’ moves. (In the game illustrated the player who goes second can always
win by making sure that in each round 3 sticks exactly are taken.).

Pages 12 and 13/Get off the island
All the numbers shown on the island can escape.
They all reduce eventually to 1. Do all numbers eventually reduce to 1?

Calculators may be useful in this investigation, as some of the numbers take many stages
to reduce to 1, but the investigation can prove very exciting and, by recording the numbers,
patterns can be detected.

Pages 14 and 15/Are you puzzled?
Turns
Corky wouldn’t have to make any turns. The Scouts should all be facing the same way they
were at the start.

The plates are numbered 5/7 and 6/3.

Animal pairs

Sum to ten
There are at least 79 groups of 10. All the numbers are used in at least one grouping.

Moving sticks

The never-never
According to the owner of the shop, Mr Zeno, the £100 would never be paid off – there would
always be a smaller fraction to pay. But in reality, after only 4 repayments, fractions of a penny
would be involved so the figures would need rounding up. It would take 13 payments to pay the
last 1p.

The 'Hole in the Wall' club
256 entered the contest.
256 > 128 > 64 > 32 > 16 > 8 > 4 > 2

Pages 16 (back cover)/501
Alice has 113 (see score board in foreground of last frame) less the single 20 she has just
scored. That’s 93 left. With some swift mental arithmetic Alice works out that the only finish
with two darts is treble 19 and then double 18. Whether her darts skill is up to it we don’t know!

It is possible to win at 501 with 3 throws of 3 darts. There are several combinations possible,
such as treble 20 seven times, treble 19 and double 12. TOP.