Upload
diana-copeland
View
220
Download
1
Tags:
Embed Size (px)
Citation preview
Why does Rudolph have a shiny nose?
A mathematical look at Christmas
Chris Budd
Can maths help Santa plan Christmas??
Santa’s problems
• How can he deliver all of the presents?
• How does he get down the Chimney?
• How does he find his way round the Earth?
• Why does Rudolph have a shiny nose?
Maths can answer all of these and …
• Helps you make great Christmas cards
• Makes Christmas magical
• Sorts out the presents in the 12 days of Christmas
• Arranges your Christmas party
Santa has 36 hours of darkness during Christmas night to deliver all of the presents
Can he get round in time?
Worlds population is 6, 000, 000, 000 people
Estimate N = 1, 000, 000, 000 homes with good children
Assume the homes are evenly distributed an average distance of H apart
H
H
But … surface area of the continents =
226,000,000,000,000 (226trillion) m2
Total area A taken up by the homes
mHNH 475000002260000000 2
Total distance that Santa has to travel = NH = 475 Gm
2NHA
Speed = 475Gm/(36*3600) = 3.6M metres per second
Sound = 375 ms-1 Light = 300 M ms-1 That’s 9600 Mach
Hyperbolic shock wave
So … why does Rudolph have a shiny nose?
Sleigh is travelling at hypersonic speeds
Air friction heats up Rudolph’s nose till it glows!
How does Santa get down the chimney?
Small diameterchimney
Large diameter
Santa
10m
beforeafter LcvL 22 /1
2mcE Solution one: Einstein’s theory of relativity
Lorentz Contraction
The faster you go the smaller you get
C = 3 00 000 000 metres per second
Quick calculation
1 000 000 000 Homes visited in 36 hours
130 micro seconds per house
Allow 1 micro second to descend a 10m chimney
Chimney velocity V = 10 000 000 metres per second
Lorentz contraction Lafter = 0.999 Lbefore is not enough
Solution two: Use a fractal
Christmas is a magical time
Maths can be part of the magic!
1 9 9 4
2 18 9 4
3 27 9 4
4 36 9 4
5 45 9 4
6 54 9 4
7 63 9 4
8 72 9 4
9 81 9 4
Orange Kangaroo
10 1 9
11 2 9
12 3 9
13 4 9
14 5 9
15 6 9
16 7 9
17 8 9
18 9 9
19 10 9
Four Aces
Great Christmas Cards
Chased Chicken Celtic Knot
A B C
Grid
Corner
PatternsCorner
Edge
Stockings and the 12 Days of Christmas
But … How Many presents did my true love send?
Day one 1
Day two 1+2
Day three 1+2+3
Day four 1+2+3+4
Day five 1+2+3+4+5
Day six 1+2+3+4+5+6
Day seven 1+2+3+4+5+6+7
Day eight 1+2+3+4+5+6+7+8
Day nine 1+2+3+4+5+6+7+8+9
Day ten 1+2+3+4+5+6+7+8+9+10
Day eleven 1+2+3+4+5+6+7+8+9+10+11
Day twelve 1+2+3+4+5+6+7+8+9+10+11+12
1 = 1
1+2 = 3
1+2+3 = 6
1+2+3+4 = 10
1+2+3+...+n = n(n+1)/2
Triangle numbersTriangle numbers
Pascal’s Triangle
Triangle numbers
Day of Christmas
Need to add them up
Use a Christmas
Stocking
364
What happened to the lost present?
OK, so my true love forgot one day
You have five friends, Annabel, Brian, Colin, Daphne, Edward
Want to invite three to a Christmas party
• Annabel hates Brian and Daphne
• Brian hates Colin and Edward
• Daphne hates Edward
Who do you invite? A C E
How to organise a Christmas parties
Now have 200 friends and want 100 to come to a partyHave a book saying who hates who Who do you invite?
Parties to check
Takes a high speed computer
Years to check them
900000000000000000000000000000000000000000000000000000000000000000
6000000000000000000000000000000000000000
Works for a party and many other problems
Using maths we can solve it in seconds
SATNAV devices … useful for Santa to find his way round the Earth!
Simulated annealing
Conclusion …. your
• Party
• Presents
• Christmas Cards
• Magic
• Visit from Santa
Are safe in the hands of a mathematician