More mathematical mysteries
Problem
Write down a three-digit number
Change the order of the digits to get a different number
Find the difference between the two three digit numbers
Add together the digits of your answer. If you arrive at a two-digit answer, add the two digits to reduce your answer to a single digit answer
Subtract $5$ from the number you have
Find that equivalent number letter in the alphabet (1=A, 2=B etc.)
Name a country that begins with letter
Using the second letter of the name of the country that you chose, name a jungle animal that starts with that letter.
Name the colour of that animal.
Using the final letter of the name of the country that you chose, name a different animal that starts with that letter.
Name a fruit that starts with the last letter of that animal's name.
Did you end up with orange or apple?
How did I know?
Student Solutions
Meg offers the following solution:
If you take any 3 digit number, and then rearrange the digits of the number and find the difference between the two numbers, the number will always be a multiple of 9.
Think about the digits as being a,b and c.
These can be rearranged and subtracted in the following ways:
abc-acb= (10b+c)-(10c+b)=9(b-c)
abc-bac=(100a+10b+c)-(100b+10a+c)=90a-90b=9(10a-10b)
abc-bca=(100a+10b+c)-(100b+10c+a)= 99a-90b-9c=9(11a-10b-c)
abc-cab=(100a+10b+c)-(100c+10a+b)=90a+9b-99c=9(10a+b-11c)
abc-cba=(100a+10b+c)-(100c+10b+a)=99a-99c=9(11a-11c)
The digits of any multiple of 9 can always be reduced to give 9. eg
18: 1+8= 9
27: 2+7 = 9
99:9+9 = 18; 1+8 = 9
981:9+8+1=18;1+8=9
So whichever numbers you use, you will always end up with 9.
9-5=4
4 translates to D using the code
The only countries beginning with D are Denmark, Djibati, Dominica and the Dominican Republic. Most people would choose Denmark.
You'd be hard-pushed to think of another jungle animal beginning with E other than elephant.
Animals beginning with K are not many in number, so Koala may come to mind.
This leads to a fruit beginning with A, which is most obviously apple.