Transcript Some Christmas Puzzles

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?
Answer
34
40
46
38
48
46
33
29
34
34
45
42
30
43
50
36
Now make up your own Christmas puzzle.
Make sure that it can be solved.
The Twelve Days of Christmas
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 ....
Fifth day ...
Sixth day ...
Seventh day ...
Eight day ...
Ninth day ...
Tenth day ...
Eleventh day ...
Twelfth day ...
Four calling birds, 3 French hens, etc.
Five gold rings, four calling birds, etc.
Six geese a-laying, five gold rings, etc.
Seven swans a-swimming, six geese a-laying, etc.
Eight maids a- milking, seven swans a-swimming.
Nine drummers drumming, etc.
Ten pipers piping, etc.
Eleven ladies dancing, etc.
Twelve lords a-leaping, etc.
Answer
How many presents did ‘my true love send to me’
on:
a)
b)
c)
the third day?
the fifth day?
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 the
twenty days in total?
Christmas Tangrams
The Tangram is a puzzle made from cutting a
square into seven pieces in the following way.
8 cm
8 cm
Answer
How many of these shapes can you make?
Which shape has the biggest area?
Which shape has the longest perimeter?
Christmas Calcogram
Santa had just finished loading the
10 + 24 realised he couldn’t
“
27689  2
me”
60 ÷ 100
67  5
3691  125
when
a single reindeer.
he said. “Where have they gone. If we don’t
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
them to send some more deer. Just then there came a
from behind the stables.
and
8  47077
319  2
Answer
First Santa saw a pair of
23596 + 33738
then two
1879  3
then at last the whole of Rudolf appeared.
“
Santa. Did you like our joke? Don’t look so worried.
8.5074 ÷11
The others are all ready but they’re hiding in a
6  9619
.”
9000 – 3492
really.”
923  5
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”
in the
He smiled. “They know you’re the
“Alright, come on, let’s 12 ÷ 20” said Santa, with a
He got out 257  2
463  8
chuckled Santa and off they went.
of relief.
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
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?
Answer
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?
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
e/
m
h²
means a × b × c
means e ÷ m
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
k – j = 22 - 20 = 2 = a
10 + 26 = 36 = r
Clue 3 = z/a
Clue 4 = 5e
z ÷ m = 52 ÷ 2 = 26 = m
Unscrambling gives…
5× e = 5 × 10 = 50 = y
Mary
Answer
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
1) Clue 1 = 4e
Clue 4 = y – f
2) Clue 1 = 5b
Clue 4 = 3e + d
3) Clue 1 = bc
Clue 4 = e(a + 1)
•
Clue 2 = d – c
Clue 3 = 7b
Clue 5 = d + e – h
Clue 2 = 2j + 2
Clue 3 = t/b
Clue 5 = 2j – 2
Clue 2 = cd – c
Clue 3 = p/a
Clue 5 = c²
Clue 6 = 2a²
Clue 7 = 2b²
Now make up your own clues for famous Christmas Characters
Grid
Designing Snowflakes
•Use isometric (triangular dotty) paper
•Your designs must have 6 lines of symmetry
•Your designs must have rotational symmetry order 6
© D Cavill
2004
Answers
Back
=1
=6
=2
=7
=3
=8
=4
=9
=5
= 10
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)
Tangram Answers
Back
Christmas Calcogram
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.
Back
35 Triangles Altogether
10
5
10
5
5
Back
Size of tray
Number of
Arrangements of Parcels
1 by 2
1
2 by 2
2
3 by 2
3
4 by 2
5
5 by 2
8
6 by 2
13
7 by 2
21
8 by 2
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
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
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
© D Cavill
2004