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The Mathematics
of MagicWarm up
Every whole number from 0 to 10 can be written as an expression using the numerals 2017 in that order using any operators. e.g. 10 = 2+ 0 + 1 + 7 How many can you find?
5 = -2 + 0 + 1 × 76 = -2 + 0 + 1 + 77 = (2 + 0)× 1 + 78 = 20 + 1 × 79 = 2 + 0 + 1 × 710 = 2 + 0 + 1 + 7
Sample Answers
0 = 2 × 0 × 1 × 71 = 20× 1 × 7
2 = 20 + 17
3 = 20 - 174 = -2 + 0 - 1 + 7
Greg CarrollSchools Outreach Officer
Australian Mathematical Sciences Institute
ChooseMaths Project [email protected]
Our PLAN for today
(P + L)(A + N)= PA + PN + LA + LN
Your PLAN has just been foiled!
1 Introduction2 AMSI3 Some Magic4 Some Maths5 Go back to 3
Discipline Profile of Mathematical Sciences – AMSI publication
Discipline Profile of Mathematical Sciences – AMSI publication
Discipline Profile of Mathematical Sciences – AMSI publication
The Mathematics
of MagicLets go!
470762837237162
2 468
762+ 237 = 999
837+ 162= 999
Still not sure?
Same front & back
The pre-prepared pad
It is important that no-one, except the person adding them, sees all the numbers.
After the 4 volunteers have written their numbers close the pad.
Turn the pad over andhand it to the fifth volunteer to add.
Your numbers My numbers
Hailstone Numbers
Hailstone Numbers
Take any number. If it is even divide by 2.If it is odd, multiply by 3 and add 1.
Why Magic?1 × 1 = 1
11 × 11 = 121111 × 111 = 12321
1111 × 1111 = 123432111111 × 11111 = 123454321
111111 × 111111 = 12456543211111111 × 1111111 = 124567654321
11111111 × 11111111 = 12456787654321111111111 × 111111111 = 1245678987654321
Why Magic?1 × 9 + 2 = 11
12 × 9 + 3 = 111123 × 9 + 4 = 1111
1234 × 9 + 5 = 1111112345 × 9 + 6 = 111111
123456 × 9 + 7 = 11111111234567 × 9 + 8 = 11111111
12345678 × 9 + 9 = 111111111
Well, not any number.
What was the chance of that?People remember that they were asked for a number between 1 and 100 , Pr = 1
100.
In reality a two digit odd number with both digits different cuts it down to Pr = 1
20.
13, 15, 17, 19 31, 35, 37, 39 51, 53, 57, 5971, 73, 75, 79 91, 93, 95, 97
You are
Magic too!
“Psychic” Testing
1. Ace through to Five of 2 contrasting suits
2. Cut as many times as the subject requests.
3. Count out 5 cards and make 2 piles.
4. Move 4 cards, it can be from either pile or a mixture, from the top to the bottom of the pile. Place the top two cards out.
5. Move 3 cards from the top to the bottom of either pile. Place the top two cards out.
6. Move 2 cards from the top to the bottom of either pile. Place the top two cards out.
7. Move 1 card from the top to the bottom of either pile. Place the top two cards out.
8. We have two cards left. Wouldn’t it be remarkable if they were the same! What is the odds of that?
So why does it work?
Investigate Clock arithmetic
Make the problem simpler by using 3 cards
Fibonacci
e.g. if we start with 3 and 5 we generate 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 . . .
Nominate two ascending numbers < 10
Find the next number by adding the previous two.
The trick for multiplying by 11.6 +3 =9
63 × 11 = 6 9 3
5 +9 =14+1 4
59 × 11 = 5 4 9 = 649
Proof BAD – Beyond All Doubt
t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
a b a + b a + 2b 2a + 3b 3a + 5b 5a + 8b 8a + 13b 13a + 21b 21a + 34b
S10 = 55a + 88b= 11(5a + 8b)= 11t7
Why Magic?Seven cards
4 red, 3 black
Why Magic?Top card under.Next card out.Repeat.
The final order is R B R B R B R.
How did they start?
• Try 5 cards r,r,r,b,b (simpler problem)• Extension get the ace to 7 of hearts spelt out in order.Answer 4, A, 6, 2, 5, 3, 7
The answer
Long DivisionDo you know your 43 number facts?4 3 438 6 8612 9 12916 12 172 Did you see how that worked?20 15 21524 18 25828 21 30132 24 34436 27 38740 30 430
https://cs4fndownloads.files.wordpress.com/2016/03/cs4fnmagicbook1.pdf
A free resource for maths magic
Choose a card, any card.Concentrate only on that card.Put your change up when you got it fixed in your mind
Predicting the lottery,“or something similar anyway”
3 volunteers
3 cards each
is 814
Add the 3
numbers
“Mathematics is a place of Magic and wonder.”
- Greg Carroll last night
Thankyou and remember
https:// www.surveymonkey.com/r/MAV17071217https://www.surveymonkey.com/r/MAV17081217