Martin Gardner, a well-respected mathematics and science writer who passed away in 2010, once posed this simple but challenging puzzle in his "Mathematical Games" column.
Express as many integers as possible, by using the digit 4 exactly four times and using
- Common mathematical functions: + − × ÷
- Concatenation and decimal points are permitted, e.g., 44, 4.4, & .4
For example,
1 = 44/44 or (4 + 4)/(4 + 4)
2 = 4/4 + 4/4
In his book Brain-Flexing Balance Problems and Other Puzzles (page 62), Ivan Moscovich writes:
It is possible to construct every digit from 1 to 10 this way. If square roots are also allowed, you can make the numbers from 11 through 20, with one exception.
Can you find the equations for the numbers from 3 to 20, and find the number that can't be expressed in this way? (There is a way to express it, but it involves a different function — can you figure it out?)
Note: Judy Ann Brown started a contest at the Math Forum (mathforum.org/yeargame/2010) with a similar challenge.
Use the digits in the current year, e.g., 2,0,1,0 (for this year) and the operations +, −, ×, ÷, ^ (raised to a power), sqrt (square root), and ! (factorial), along with grouping symbols, to write expressions for the counting numbers 1 through 100. This year the contest will also allow the use of decimal points and double-digit numbers. This year we'd also like to challenge you to try to use the digits in the order 2, 0, 1, 0 in your expressions. We'll accept digits in any order, but we're more likely to publish an expression like 2 + 0 - 1 + 0 than 2 * 0 + 0 + 1. Teachers may print out worksheets for students to record their findings, or may print sheets of manipulatives for students to use.
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