

[Photo by Amanda M Hatfield.]

Have you made a resolution to exercise your mental muscles this year? Then please join us for the 2009 Mathematics Game. Here are the rules:

Use the digits in the year 2009 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, 9 may be used.

The decimal point may be used, as in .9, .02, etc.

Multi-digit numbers such as 29 or 902 may be used, but preference is given to solutions that avoid them.

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.]

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 two years. I’m looking forward to the fun!

Clarifying the Rules

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 order, using the 2 first and the 9 last.

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, 9.

“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, 9.

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!).”

Also, 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.

Keeping Track

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) here:

Percent solved = 90%. Wow!

Numbers that remain unsolved =

52, 67-68, 74, 76-78, 86, 97-98.

Depending on how busy life gets, the tally may lag a few days behind the results posted in the comments, so be sure to scroll down and read the latest news.

And if you would like to join me in the “extended edition” game…

Still looking for an expression that does NOT use multi-digit numbers:

57, 84-85, 87-89, and 91-95.

57, 84-85, 87-89, and 91-95. Found a way to keep the digits in order: (62%)

1-36, 38, 40, 42, 44-46, 48, 50, 54, 57, 60, 62-64, 66, 69, 71-73, 75, 80-81, 90, 96, 99-100.

Related Number Puzzles

Check out Happy 41*7^2, What is interesting about the number 2009?, and The number 2009.

You may also enjoy the Famous Four 4’s. Or have a look at All the King’s Digits, or try Jonathan’s 4, 3, 2, 1 Puzzle.

This post is featured in the Carnival of Homeschooling: Week 158.

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and you’ll be among the first to hear about new books, revisions, and sales or other promotions.