Mind Reading
1. Think of a 2-digit natural number (a positive whole number).
2. Get the sum of the 2 digits.
3. Subtract the sum of the 2 digits from the original number.
4. Find the result in the table below.
5. Look at the shape that is in the same cell as your number.
6. Concentrate hard on that shape.
You were thinking about a ____________
Want to know how this is done? Click on www.themathstutor.ie/mind-reading.html
You were thinking about a TRIANGLE! (If not, then please check your arithmetic!)
Explanation 1: using Arithmetic
You can take it on trust that this works every time, but it’s nice to know why.
Let’s figure it out by playing around with it and trying a few examples:
Original Number
Sum of Digits
Subtracting:
43
7
36
12
3
9
77
14
63
51
6
45
10
1
9
98
17
81
33
6
27
59
14
45
What do you notice about the results?
They are all multiples of 9. Interesting! I wonder why? Let’s explore further.
This time, let’s subtract the digits individually (instead of summing them first and then
subtracting that). It amounts to the same thing.
Original Number
Second Digit
Subtracting:
First Digit
Subtracting:
43
3
40
4
36
12
2
10
1
9
77
7
70
7
63
51
1
50
5
45
10
0
10
1
9
98
8
90
9
81
33
3
30
3
27
59
9
50
5
45
Notice that when we subtract the second digit we always get a multiple of 10 (a digit with a
0 after it). In fact it is 10 times the first digit e.g. 40 is 10 times 4.
When we subtract the first digit, we are taking away one tenth of the value of the number,
so we are left with 9/10ths of the value. So that has to be a multiple of 9. For example, 40
take away 4 leaves 36, which is 9 times 4.
So, we can see using arithmetic that our final answer will always be a multiple of 9.
That’s why everyone was looking at some multiple of 9 (but not necessarily the same one!).
If you look again at the table, you can see that all multiples of 9 are in cells with a triangle
(sneaky!).
So that’s how we knew that everyone was looking at a triangle.
Now that you understand how it works, try it out on a friend!
Explanation 2: using Algebra
We can also use algebra to give a more elegant explanation:
Think about the original number.
Its value is (10 times its first digit + its second digit).
Let’s represent the first digit with the variable a and the second digit with the variable b.
So we could represent the number as (10a + b)
The sum of the 2 digits can be represented as (a + b)
Subtracting the digits gives
(10a + b) - (a + b)
= 10a + b – a – b
=10a – a + b – b
=9a
But of course 9a is always a multiple of 9.
So everyone was looking at some multiple of 9 (but not necessarily the same one!).
If you look again at the table, you can see that all multiples of 9 are in cells with a triangle
(sneaky!).
So that’s how we knew that everyone was looking at a triangle.
Now that you understand how it works, try it out on a friend!
If you need any more help in understanding this, feel free to get in touch with us!
Source:
http://www.cut-the-knot.org/Curriculum/Magic/MindReaderNine.shtml