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Can you make the numbers 1 to 20 using only the digit '4' and the operations +, -, ÷ and ×?

For example

• 1 = 4÷4
• 11 = 44÷4
• 19 = 4×(4+4)-4-4-4-(4÷4)

Then, you might just win Dr Mike's "Fours" Contest!

On August 20, 2009, this website was four years old. To celebrate, I organized a free contest. It's based on the famous puzzle that inspired this game, where you have to make as many numbers as possible using the digit '4' four times.

However, some numbers don't need four 4's, and others need more - maybe five, six, seven or more 4's.

The idea for this contest was to make as many numbers as possible, using as few 4's as possible. The more numbers you make, and the fewer 4's you use, the better your chance of winning!

To enter, you had to get your contest form in the post by October 20, 2009. I'll give a week or so for the results to get in, then post out nice certificates to the winners in each grade (as well as announce them here). However, if you like the sound of this contest, but it's too late to enter, why not organize a contest within your school or class? I'll post ideas for how to do that later.

These were the steps to enter the contest :

• Step 1 : Print out, and fill up the entry form
• Step 2 : Print out as many answer pages as you want.
• Step 3 : Start with page 1 of the answer sheet, make each of the numbers 1 to 20 using only the digit 4, and the operations +, -, ÷ and ×. Use as few 4's as you can, and check each answer carefully.
• Step 4 : Then, do the same with page 2, then page 3, page 4 and so on. There's no limit to the number of pages you can do.
• Step 5 : Write your name on each page, and how many 4's you used.
• Step 6 : Clip or staple your pages together with the entry form, and send them all to the address on the entry form. Be sure to post it before the 20th of October 2009!

When I've collected all the entries, I'll decide the winner as follows :

• First, I'll find the entries using the fewest number of 4's on answer sheet page 1. The winner will be one of these.
• I expect this will result in a tie - to break the tie, I'll find the entry with the fewest number of 4's on answer sheet page 2. What if you didn't do page 2? Too bad!
• If there's still a tie, I'll look at page 3, then page 4 and so on - so it's important to carefully find the best answers you can on as many pages as possible, but especially the early pages!