© Juliet & Charles Snape 2011
Volume 16
CIRCA 47: Solutions

Front cover/Zasra's adder
It doesn’t matter what number is
picked, if it’s more than 2 the counting
will always end on the ‘eye’ symbol.

This is how it works ... The neck takes
away 2 from the number you chose.
You then count anticlockwise the
remainder of your number.Finally,
you count the full number clockwise
but instead of going back down the
neck you count on a further 3 around
the snake, because you do not count
the symbol you landed on, nor the
two on the neck. This brings you to
the symbol of the ‘eye’.


Pages 2 and 3/The amazing Fibonacci numbers

The Fibonacci numbers are made by adding the previous two numbers to generate the next. The sequence continues:

21, 34, 55, 89, ...

There would be 21 pairs of rabbits by August.

Growth spirals that are pairs of Fibonacci numbers occur in celery, cactus plants, pine cones, palm trees, daisies, sunflowers and many more plants.

Note: Readers may not have noticed the curious property of the magician’s name, Robert Trebor. It is a palindrome, that is, it reads the same backwards as forwards.

Pages 4 and 5/Zasra's amazing arithmetic

1089 (page 6)
This trick works for all 3-digit numbers except palindromes (numbers that read the same backwards or forwards e.g. 727). However, these are excluded as Zasra says all 3 digits must be different.

Multiplying 1089 by the numbers 1 to 9 produces some interesting patterns. What can you find in this table?

1 x 1089 = 1089
2 x 1089 = 2178
3 x 1089 = 3267
4 x 1089 = 4356
5 x 1089 = 5445
6 x 1089 = 6534
7 x 1089 = 7623
8 x 1089 = 8712
9 x 1089 = 9801

2-digit trick (page 7)
The sum of the digits in all 2-digit numbers subtracted from the original 2-digit number always results in 9 or a multiple of 9. If you take a close look at the symbol table you will see that all the multiples of 9 (18, 27, 36, ...) have the same symbol. The symbols for the other numbers are
just randomly spread.

What happens if you do the trick with 3-digit numbers?

Pages 6 and 7/Number patterns

Correct answers will spell out the word MAGIC.

The investigation of triangle numbers will show that two consecutive numbers in the pattern sum to a square number (the generating rule of the silk scarves).

An investigation into the digital routes of the number pattern of ‘add three’ will show a repeat of 3, 6, 9. What happens with other times tables?

Page 8 and 9/Magic pattern

The props show the numbers: 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0.

The rule is “if odd, subtract 3; if even, add 1”. (Does this rule always produce alternating Odd and Even numbers?)

There are only three false clues in rooms that would break the generating rule.

If the rule is perceived as being independent of whether the numbers are odd or even (i.e. just subtract 3, then add 1, regardless of what the first number is), further investigation will discover that any odd start number will reach zero and any even starting number will reach 2.

Page 10 and 11/Magic squares

The difference between the Lo Shu and the Kubera-Kolam is that each cell has had 19 added to it so that the summation of the square becomes 72. (The digital route of 72 is 9, which Hindus considered a divine number.) Essentially it is the same as the Lo Shu as there is only one arrangement that works in a 3 x 3 square (other percieved solutions are reflections or rotations).

The 4 x 4 square offers more opportunity for arrangements and the method shown illustrates a way of doing one. Comparing it to Dürer’s square will help understand how. Changing the summation is also interesting; the summation of a 4 x 4 square with numbers 1 - 6 is 34, so to make a summation of 50,
4 must be added to each cell. investigate which numbers work best.

Page 14 and 15/Are you puzzled?

Page 12 and 13/Great escape mystery

Ricky is escaping from a sack.

Frieda / Theatre Royal / Barrel
Ricky / Children's Birthday Party / Sack
Eric / Circus / Water Tank
Harry / Village Hall / Suitcase
Letty / TV Studio / Safe
Walter / End of the Pier Show / Chest

Solving the mystery (this is one way, there are others): Going through the clues without too much puzzling you can fill in all the headings on the grid (who?, where?, what?). From clue 4 we learn that Harry is performing in the village hall and clue 7 tells us that Harry is escaping from a suitcase.
Ticks and crosses are added as appropriate. Clue 9 eliminates Frieda from the TV studio and the circus.
She’s in a barrel at the theatre. Clue 5 tells us that Walter is at the end of the pier. From clue 3 we
know that Eric is not in the TV studio. That leaves only the circus as his venue and the water tank as
the thing he is escaping from. Letty, thus, can only be at the TV studio where she is escaping from
the safe (clue 9). This leaves just one empty box, from this we can deduce that Ricky is escaping
from a sack.

For information: Escapologists use a variety of methods for escaping; these include regurgitating swallowed keys and lock picks, chest enlargement through breathing and dislocating shoulders.
Not to be tried at home!

1. Where is the five of clubs?
(D) is 5 of Clubs.

2. Trick cuts
The third piece of rope is 2 metres long.

3. Which is the match?
(6) is the matching silhouette.

4. Complete the grid
There are 13 triangles.

Alice: Button box(back page)
There are 3 red buttons in the box.

Although the problem can be solved with algebra it is a simple enough problem to be sorted out by arithmetic. A single red button set would contain 24 buttons (1 red, 4 grey, 5 green, 6 blue 8 yellow). 72 ÷ 24 = 3.

Page 15/Shading squares
The results below show the possible shadings for shapes with 1 to 6 cells (they include all cells unshaded and
all cells shaded).