Ab 24. The Christmas Puzzle The pictures in the grid each represent a number between 1 and 10. Using...

a b

2 4

The Christmas Puzzle

The pictures in the grid each represent a number between 1 and 10.

Using the totals of each row and column, can you work out what the value of each shape is?

4534

48

43

33

36

29

34

46

38

42

40

46

34 30 50

Answer

Now make up your own Christmas puzzle.

Make sure that it can be solved.

On the first day of Christmas my true love sent to me:A partridge in a pear tree.

On the second day of Christmas my true love sent to me:Two turtle doves and a partridge in a pear tree.

On the third day of Christmas my true love sent to me:Three French hens, two turtle doves and a partridge in a pear tree.

Forth day .... Four calling birds, 3 French hens, etc.Fifth day ... Five gold rings, four calling birds, etc.Sixth day ... Six geese a-laying, five gold rings, etc.Seventh day ... Seven swans a-swimming, six geese a-laying, etc.Eight day ... Eight maids a- milking, seven swans a-swimming.Ninth day ... Nine drummers drumming, etc.Tenth day ... Ten pipers piping, etc.Eleventh day ... Eleven ladies dancing, etc.Twelfth day ... Twelve lords a-leaping, etc.

The Twelve Days of Christmas

How many presents did ‘my true love send to me’

on:

a) the third day?b) the fifth day?c) the tenth day?

How many presents did ‘my true love send to me’ over the twelve days in total?

What happens if there were 20 days of Christmas instead of twelve?

How many presents would ‘my true love sent to me’ over thetwenty days in total?

Answer

Christmas Tangrams

The Tangram is a puzzle made from cutting a

square into seven pieces in the following way.

8 cm

8 cm

How many of these shapes can you make?

Which shape has the biggest area? Which shape has the longest perimeter?

Answer

Santa had just finished loading the 3691 125 when

10 + 24 realised he couldn’t 67 5 a single reindeer.

“ 27689 2 me” he said. “Where have they gone. If we don’t

60 ÷ 100 soon we’ll never get round all the children

tonight.”

He grabbed 900 – 386 2396 + 5343 and rang it loudly.

He waited and listened, but nothing happened. “Come on boys”, he muttered, or

1191 3 I’ll have to phone the 2 ÷ 100 and 319 2

them to send some more deer. Just then there came a 8 47077

from behind the stables.

Christmas Calcogram

First Santa saw a pair of 23596 + 33738 then two

1879 3 then at last the whole of Rudolf appeared.

“ 8.5074 ÷11 Santa. Did you like our joke? Don’t look so worried.

The others are all ready but they’re hiding in a 463 8 in the6 9619 .” He smiled. “They know you’re the

9000 – 3492 really.”

“Alright, come on, let’s 12 ÷ 20” said Santa, with a 923 5 of relief.

He got out 257 2 list. “Now let’s see, I’ve got a computer

for 28867 11 and the 637 ÷ 1000

for Gemma, and the bike for 3 246017.”

“1.0101 0.4” chuckled Santa and off they went.

Answer

The Christmas Star

Here is a Christmas Star

How many triangles can you find?

Show where the triangles are as clearly as possible.

You will need to draw more than 1 diagram

Answer

The Christmas Star

Here is a more complicated Christmas Star

How many triangles can you find?

Show where the triangles are as clearly as possible.

You will need to draw more than 1 diagram

Packing

The elves in one of Santa’s workshops always pack identically sized parcels (2 units by 1 unit) into trays which are 2 units wide but which have different lengths.

How many different ways can the parcels be packed into a 3 by 2 tray?

How many different ways can the parcels be packed into a 3 by 2 tray? 3 Ways

How many different ways can the parcels be packed into a 4 by 2 tray?

Investigate how many different ways the parcels can be packed into trays 2 units wide but with different lengths?

Can you find a general rule?

Answer

• How many arrangements are there for a 20 by 2 tray?

Extension

•What happens if the tray is always 3 units wide?

Answer

Solve the algebraic clues to find the letters in the name of a Christmas character.

You will need to put the letters in the correct order.

Rules of algebra

a + b means the value of a added to the value of b

c – d means take away the value of d from the

value of c

3x means 3 times the value of x

3x = 3 × x

pq means p × q

abc means a × b × c

e/m means e ÷ m

h² means h × h ( h squared)

Name the Christmas Character

a b c d e f g h i j k l m

2 4 6 8 10 12 14 16 18 20 22 24 26

n o p q r s t u v w x y z

28 30 32 34 36 38 40 42 44 46 48 50 52

Clue 1 = k - j Clue 2 = e + m

Clue 3 = z/a Clue 4 = 5e

k – j = 22 - 20 = 2 = a 10 + 26 = 36 = r

z ÷ m = 52 ÷ 2 = 26 = m 5× e = 5 × 10 = 50 = y

Unscrambling gives… Mary

1) Clue 1 = 4e Clue 2 = d – c Clue 3 = 7b

Clue 4 = y – f Clue 5 = d + e – h

2) Clue 1 = 5b Clue 2 = 2j + 2 Clue 3 = t/b

Clue 4 = 3e + d Clue 5 = 2j – 2

3) Clue 1 = bc Clue 2 = cd – c Clue 3 = p/a

Clue 4 = e(a + 1) Clue 5 = c² Clue 6 = 2a²

Clue 7 = 2b²• Now make up your own clues for famous Christmas Characters

a b c d e f g h i j k l m

2 4 6 8 10 12 14 16 18 20 22 24 26

n o p q r s t u v w x y z

28 30 32 34 36 38 40 42 44 46 48 50 52

Answer

Designing Snowflakes

•Use isometric (triangular dotty) paper

•Your designs must have 6 lines of symmetry

•Your designs must have rotational symmetry order 6

Grid

© D Cavill

2004

Answers

= 1

= 3

= 2

= 4

= 5 = 10

= 9

= 8

= 7

= 6

Back

The Twelve Days of Christmas

On the third day, 6 presents were sent.

On the fifth day, 15 presents were sent.

On the tenth day, 55 presents were sent.

Altogether, over the twelve days

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 presents

Back

• Note that the number of presents given each day is a triangular number and the sum of the first n triangular numbers is called a tetrahedron number.

Hence 364 is a tetrahedron number.

• Over 20 days the total number of presents would be:

364 + 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 = 1540.

• A general formula to give the number of presents on day n (triangular numbers)

½ n (n +1) • A general formula to give the number of presents sent altogether

over the first n days is:

1/6 n³ + ½ n² + 1/3 n = 1/6 n (n + 1)(n + 2)

Back

Tangram Answers Back

Santa had just finished loading the SLEIGH when HE realised he couldn’t SEE a single reindeer.

“BLESS me” he said. “Where have they gone. If we don’t GO soon we’ll never get round all the children

tonight.”

He grabbed HIS BELL and rang it loudly.

He waited and listened, but nothing happened. “Come on boys”, he muttured, or ELSE I’ll have to phone

the ZOO and BEG them to send some more deer. Just then there came a GIGGLE from behind the

stables. First Santa saw a pair of HEELS then two LEGS then at last the whole of Rudolf appeared.

“HELLO Santa. Did you like our joke? Don’t look so worried.

The others are all ready but they’re hiding in a HOLE in the HILLS” He smiled.

“They know you’re the BOSS, really.”

“Alright, come on, let’s GO ” said Santa, with a SIGH of relief. He got out HIS list.

“Now let’s see, I’ve got a computer for LESLIE and the LEGO for Gemma, and the bike for

ISOBEL.”

“HOHOHO” chuckled Santa and off they went.

Christmas Calcogram Back

5

55

10

10

35 Triangles Altogether Back

Size of tray Number ofArrangements of Parcels

1 by 2

2 by 2

3 by 2

4 by 2

5 by 2

6 by 2

7 by 2

8 by 2

1

2

3

5

8

13

21

34

Back

To find out how many ways there are to pact a 7 by 2 tray, add together the number of ways to pack a 5 by 2 tray and a 6 by 2 tray

To find out how many ways there are to pact a n by 2 tray, add together the number of ways to pack a n-1 by 2 tray and a n-2 by 2 tray

Wn = Wn-1 +Wn-2

Ways to pack a n by 2 tray

Back

1235813213455891442333776109871597258441816765

There are 10946 Ways to pack a 20 by 2 tray!

This number sequence is called the Fibonacci Sequence after one of the

greatest European mathematicians of the middle ages, Leonardo of Pisa, better

known as Fibonacci.

Back

Name the Christmas Character

1) Santa

2) Jesus

3) Rudolph

Back

Back

© D Cavill

2004