Is it Magic or is it Maths?
Problem
Here are three 'tricks' to amaze your friends.
But the really clever trick is explaining to them why these 'tricks' are maths and not magic. Like all good magicians, you should practise by trying them. Can you explain how they work?
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This trick will impress even your maths teacher. 
 Think of a number.
 Double it.
 Add $10$.
 Halve it.
 Take away you original number.
 Is your answer $5$?
Try this with a different
starting number. Did you get a different result? Why does this
happen?
Write the answer on a piece of paper without letting anybody see it
and seal it in an envelope. Have somebody hold the envelope and at
the end ask them to open it and reveal the number you wrote at the
beginning. Wow, Magic!
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Guess how much money people have in their pockets! 
Without giving you any information, ask a friend to count the value of some coins and write the amount on a piece of paper. Then ask your friend to:
 Double the amount.
 Add the first odd prime number to the new total.
 Multiply the result by 1/4 of 20.
 Subtract the lowest common multiple of 2 and 3.
For the grand finale, you ask for the final answer. Take off the last digit and you will be able to work out how much the coins are worth!
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Amaze your audience by working out not only their age but also
what size shoe they wear! Wow them even more by telling them how
the maths works.
Give them the following directions but tell them not to show you any calculations: 
 Write down your age.
 Multiply it by 1/5 of 100.
 Add on today's date (e.g. 2 if it's the 2nd of the month).
 Multiply by 20% of 25.
 Now add on your shoe size (if it's a half size round to a whole number).
 Finally subtract 5 times today's date.
 Show me you final answer!
Look at the answer, the
hundreds are the age and the remaining digits are the shoe size. If
for instance somebody shows you 1105, there are 11 hundreds  the
age, and the remaining digits 05 (or 5) show the shoe size.
Now, how on earth does that work?
Student Solutions
Jannis and Rohaan from Long Bay Primary School sent us the following explanation:
1. First we need to make one change to make the whole problem ridiculously easy.First we noticed, you have 'double it' and 'halve it' this brings you back to your original number, but as you have 'add ten' in the middle of 'double it' and 'halve it' you must add five because 'add ten' is before 'halve it'.
So now we have:
 Think of a number
 Add 5
 Take away you original number
 Is your answer 5?
As you can see you are just adding five to your original number then taking away your original number which is five.
2. First we need to make a few changes to simplify the whole problem.
'Double the amount' and 'Multiply the result by 1/4 of 20' effectively is 'multiply by 10'.
But since 'Add the first odd prime number to the new total' is before 'Multiply the result by 1/4 of 20' you just need to multiply the first odd prime number by 5 (3 times 5 is 15).
But since you have to subtract 6 (the lowest common multiple of 2 and 3) you may as well add 9 instead of 15 (15 6 = 9).
So now we have:
 Times it by 10
 Add 9
So then when you take off the last digit you are effectively subtracting 9.
What you are left with is 10 times the original number.
3. To explain it, you must first take off 'Add on today's date' and 'Finally subtract 5 times today's date'.
This is because after you add on the date you multiply it by 5 and you then 'subtract 5 times today's date'.
Then you may as well also change 'Multiply it by 1/5 of 100' and 'Multiply by 20% of 25' to 'multiply it by 100' because it is effectively the same thing as above.
So now you have:
 Write down your age
 Multiply it by 100
 Now add on your shoe size.
This effectively gives you two places after your age to add on your shoe size.
Gabriel Tan from Catholic High School used algebra to explain how these tricks work:
For the first trick:
Let x be the number my friend was thinking of.
1. x .........................Think of a number
2. 2x ........................Double it
3. 2x+10 .................Add 10
4. x+5 ......................Halve it
5. 5 .........................Take away your original number
And there you have it  5.
For the second trick:
Let x be the amount of money my friend has.
1. x ...............................Double the amount
2. 2x+3 .........................Add the fist odd prime number (3) to the new total
3. 10x+15 .....................Multiply the result by 1/4 of 20
4. 10x+9 .......................Subtract the lowest common multiple of 2 and 3
5. 10x ...........................Take off the last digit
And there you have it  the amount x.
For the third trick:
Let x be my friend'??s age, y be today'??s date, and z be the shoe size.
1. x ..............................Write down your age
2. 20x ..........................Multiply it by 1/5 of 100 (20)
3. 20x+y ......................Add on today's date
4. 100x+5y ..................Multiply by 20% of 25 (5)
5.100x+5y+z ...............Now add on your shoe size
6. 100x+z ....................Finally subtract 5 times today's date
And there you have it  the amounts x and z.