### Pyramids

What are the missing numbers in the pyramids?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

# Is it Magic or Is it Maths?

##### Stage: 3 Challenge Level:

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
• Take away you original number

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

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:
• Multiply it by 100

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
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)