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

particular solution please e-mail).

**Volume ****5 (issues 13, 14 and 15)**

**CIRCA 13: Solutions**

**Front cover/Starship maze and Cosmic maze**:

Please e-mail for solutions

**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.

particular solution please e-mail).

Please e-mail for solutions

Both contain a transposition error.

Chocolate should be 5 023247 047329 and

Beans should be 5 000157 024886.

Hidden message: YOU HAVE FOUND THE GOLD

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;.

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

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

A, B, E and F are possible (there are several routes for each one.

C and D cannot be done.

Spiral to square

Please e-mail for solution.

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.

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.

of the magazine. (This will work for any pair of single-digit numbers.).

numbers. For an algebraic explanation of why this works see below (answers for page 14).

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.

Best answers:

Fraction A, C and E

Decimal D, H and I

Percentages B,F and G

80%. (The image needs to be one fifth smaller.)

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.

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.

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

Zasra asks you to multiply your number by 3...

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.

were they had come from. The other directions were then correct. TOP

The least number of moves needed is 15.

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

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

(d) Tunnel = 7 Spaceship = 6

No other combination is possible.

Please e-mail for a solution.

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.).

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.

Corky wouldn’t have to make any turns. The Scouts should all be facing the same way they

were at the start.

Add up

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

Animal pairs

Please e-mail for a solution.

Sum to ten

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

Moving sticks

Please e-mail for a solution.

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

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.