[Photo from Wikipedia.]
So grab a partner, slip into your workout clothes, and pump up those mental muscles!
Here are the rules:
Use the digits in the year 2011 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 1.
- 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.
You may use multifactorials:
- (n!)! = a factorial of a factorial, which is not the same as a multifactorial
- n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n
- n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3
[Note to teachers: The bonus rules are not part of the Math Forum guidelines. They make a significant difference in the number of possible solutions, however, and they 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 other busy plans for the weekend, so this tally will lag a few days behind the results posted in the comments.
Percent confirmed = 97%.
Reported but not confirmed =
Numbers we are missing =
And if you would like to join me in the “extended edition” game…
Made it with Math Forum rules = 54%:
1-16, 18-26, 29-33, 36, 40, 42, 45, 49-51, 54-56, 59-61, 64, 66, 70-73, 80-81, 98-100.
Found an expression *without* multi-digit numbers:
1-33, 35-41, 43-55, 57-65, 70-75, 78-85, 87-92, 94-100.
Found a way to keep the digits in order:
1-33, 35-41, 43-55, 57, 58-68, 70-75, 79-81, 83, 88, 90, 92, 94-100.
Math Forum will begin publishing student solutions after February 1, 2011. Remember, you may not submit answers with multifactorials or repeating decimals to the Math Forum site.
- 2011 Mathematics Game Worksheet
For keeping track of which numbers you’ve solved.
- 2011 Mathematics Game Manipulatives
This may help visual or hands-on thinkers.
- 2011 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 Do’s and Don’ts
Finally, here are a few rules that players have found confusing in past years.
These things are allowed:
- 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 2-0-1-1 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, 1.
- The multifactorial = the product of all integers from 1 to n that are equal to n mod k. The double factorial and triple factorial may be written as !! and !!!, respectively, but for higher multifactorials BOTH n and k must be constructed from the year digits,
These things are not allowed:
- “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, 1.
- 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.
With only three distinct digits to work with this year, you will have to use every trick in the book to create variety in your numbers. Especially, you will want to play around with decimals, double-digit numbers, and factorials of all sorts. Remember also that dividing or using a negative exponent creates the reciprocal of a fraction, which can flip the denominator up where it can be more helpful.
For more tips, check out this comment from the 2008 game.
Heiner Marxen has compiled hints, links, and results for past years (and for the related Four 4’s puzzle). Dave Rusin describes a related card game, Krypto, which is much like my Target Number game. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.