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