## Mathematical Structures

Competition Autumn 2012

Notice
I didn't make it clear that all digits should be used.
So there will be two competitions:
1. in one of them all digits must be used;
2. in the other you can use as few as you need, but no digit
can be used more than once.
The winner of each competition will get a prize.

One of the students on the module challenged me to make the number 100 from the digits 1,2,3,...,9 together with mathematical operations.

I have decided to add a twist to the problem and to offer a prize.

#### The problem

You have to make the number 100 using only the digits 1,2,3,...,9 and standard mathematical operations.

The digits are free but you have to pay for the mathematical operations you use. The rules are:

• The arithmetic operations +, −, ×, ÷ (or / ) cost £2 each.
• A pair of brackets costs £4.
• Raising to a power and extracting a root cost £6.
• Juxtaposition of digits costs £8. Thus, writing 147 in your formula would cost £16 since there are two juxtapositions.
• Other standard mathematical operations represented by symbols, such as factorial, floor, etc., cost £10. (The floor function is represented by a pair of brackets; the £10 charge buys you both brackets.)
• Mathematical functions represented by words, such as tan or log, cost £12.
• The winning formula will be the one whose cost is smallest. The prize will be the cost of the winning formula. In the case of a tie, the editor will resolve the issue by randomization (that is, draw one entry out of a hat).
• The closing date for the competition is Friday, 2 November 2012.
• The editor's decision is final, but correspondence will happily be entered into.