Are you ready for a challenge? Join us for the 2008 Mathematics Game. Here are the rules:
Use the digits in the year 2008 and the operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), and ! (factorial) — along with parentheses, brackets, or other grouping symbols — to write expressions for the counting numbers 1 through 100.
- All four digits must be used in each expression.
- Only the digits 2, 0, 0, 8 may be used.
- Multi-digit numbers such as 20, 208, or .02 MAY be used this year.
- The square function may NOT be used.
- The integer function may NOT be used.
[See Dr. Math’s Why does 0 factorial equal 1?]
For this game we will accept the value:
[See the Dr. Math FAQ 0 to the 0 power.]
According to the Math Forum 2007 game page:
For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. This year may prove to be a challenge.
This year, also — those zeros make it tricky! Still, working together, we managed to find 76 answers last year.
Use the comments section below to post a running list of the numbers you have been able to calculate. We had quite a lively discussion last year. I will be out of town this weekend, visiting the in-laws for a belated Christmas celebration, but I hope to dig into this puzzle during the long drive. I look forward to seeing your comments when I get back….
Warning: Some teachers may be using this puzzle as a classroom assignment, and there will always be students looking for people to do their work for them. Please do not post any solutions!
Edited to Add
Many people have asked in the comments, so I will keep a running tally here. These are our results so far:
Percentage solved = 86%.
Numbers that remain unsolved =
43, 52, 53, 67, 68, 70, 72, 76, 87, 92, 93, 94, 98, 99.
Clarifying The Rules
The only digits that can be used to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 0, 8.
- “0!” is not a digit, so it cannot used to create a base-10 numeral.
- The decimal point is not an operation that can be applied to other mathematical expressions: “.0!” does not make sense.
No exponent may be used except that which is made from the digits 2, 0, 0, 8.
- You may not use a square function, but you may use “^2.”
- You may not use a cube function, but you may use “^(2+0!).”
- You may not use a reciprocal function, but you may use “^(-0!).”