[Photo by pfala.]
Here are the rules:
Use the digits in the year 2010 to write mathematical expressions for the counting numbers 1 through 100.
- All four digits must be used in each expression. You may not use any other numbers except 2, 0, 1, and 0.
- You may use the arithmetic operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), and ! (factorial). You may also use parentheses, brackets, or other grouping symbols.
- You may use a decimal point to create numbers such as .1, .02, etc.
- Multi-digit numbers such as 20 or 102 may be used, but preference is given to solutions that avoid them.
You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
[Note to teachers: This rule is not part of the Math Forum guidelines. It makes a significant difference in the number of possible solutions, however, and it should not be too difficult for high school students or advanced middle schoolers.]
How Does It Work?
Use the comments section below to post a running list of the numbers you have been able to calculate. You may also share relatively cryptic tips and hints, but be warned: Some teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their work for them.
Do not post your solutions. I will delete them.
I know of no authoritative list of numbers that can be made with each year’s digits, so we will rely on our collective wisdom to decide when the game is done. We had a lively discussion the last few years. I’m looking forward to the fun!
As the game results are reported below, I will keep a running tally of confirmed results (that is, numbers reported by two or more players). Today is Kitten’s birthday, however, and we have company coming in for the weekend, so this tally will lag a few days behind the results posted in the comments.
Percent confirmed = 74%.
Numbers we are missing =
34, 38, 43, 47, 52, 56, 58, 62, 66-69, 74, 76-78, 84-87, 89, 91, 93-96.
And if you would like to join me in the “extended edition” game…
Found an expression *without* multi-digit numbers:
1-13, 15-33, 35-37, 39-41, 44-46, 48-51, 53-54, 59-61, 63-65, 70-73, 75, 79-83, 90, 99 and 100.
Found a way to keep the digits in order:
1-13, 16, 18-26, 27-33, 36, 40, 49-51, 53-54, 59-61, 64, 70-73, 88, 92 and 97.
Update: Math Forum has posted their 2010 Student Solutions page. Since they did not allow repeating decimals, their list of solutions is a bit shorter than ours.
- 2010 Mathematics Game Worksheet
For keeping track of which numbers you’ve solved.
- 2010 Mathematics Game Manipulatives
This may help visual or hands-on thinkers.
- 2010 Mathematics Game Student Submissions Information
For elementary through high school students who wish to share their solutions, to be posted beginning February 1st.
Clarifying the Rules
Finally, here are a few things that some players have found confusing in past years:
- By definition: . [See Dr. Math’s Why does 0 factorial equal 1?]
- For this game we will accept: . [See the Dr. Math FAQ 0 to the 0 power.]
- Unary negatives are allowed. That is, you may use a “-” sign to create a negative number. This is particularly helpful if you are trying to keep the digits in 2-0-1-0 order.
- The only digits that can be used to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 1, 0.
- “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, 1, 0.
- 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!).”
- You have to “hit” each number from 1 to 100 exactly — no rounding off or truncating decimals allowed. You may not use the integer function.
For more hints, check out this comment from the 2008 game.