2020 Mathematics Game — Join the Fun!

New Year’s Day

Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual.

Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time.

However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion.

— Mark Twain
Letter to Virginia City Territorial Enterprise, Jan. 1863
quoted in Early Tales & Sketches, Vol. 1: 1851-1864 (affiliate link)

If you’d like to enjoy a mathematical New Year’s Resolution, may I recommend Evelyn Lamb’s Math Reading Challenge? I haven’t decided if I’m going to follow along, but it does look like fun.

Meanwhile, I do resolve to challenge myself with more math puzzles this year. Would you like to join me?

Here’s a great way to start: with the 2020 Mathematics Game!

Rules of the Game

Use the digits in the year 2020 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

• You must use all four digits: 2, 0, 2, and 0. You may not use any other numbers.

• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.

• You may use a decimal point to create numbers such as .2, or you may create multi-digit numbers such as .02 or 20 or 202.

My Special Variations on the Rules

• Challenge yourself: Keep the year digits in 2-0-2-0 order, if you can. And stick to the single-digit numbers as long as possible, leaving multi-digit numbers like .02 or 20 as a last resort.

• You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: You can’t submit answers with repeating decimals to Math Forum.

• And this year, you may also use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I feel much more creative when I can wrangle a solution without invoking double factorials, but this year we’ll need all the help we can get.

How To Play

With only two different digits to use, we will need plenty of arithmetic tricks to create variety in our numbers. Experiment with decimals, two-digit numbers, and factorials. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it might be more helpful.

• Use the comments section below to share the numbers you find.

• But please don’t spoil the game by telling us how you made them!

You may give relatively cryptic hints, especially for the more difficult numbers, but be careful. Many teachers use this puzzle as a classroom or extra-credit assignment, and there will always be students looking for people to do their homework for them.

• Do not post your calculations. I will delete them.

We’ve had some lively discussions in past years. I’m looking forward to this year’s fun!

Keeping Score

As players report their game results below, I will keep a running tally of confirmed results (numbers reported found by two or more players).

Percent confirmed: 0%
Reported but not confirmed: 0.

My family will be traveling to see extended family and then welcoming our expat daughter for a visit home. (Hooray!) So this tally will almost certainly lag behind the results posted in the comments.

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done.

Students in 1st-12th grade may wish to submit their answers to the Math Forum, which will begin publishing student solutions after February 1, 2020. Remember, Math Forum allows double factorials but will not accept answers with repeating decimals.

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:

• You must use each of the digits 2, 0, 2, 0 exactly once in each expression.

• For this game, 0! = 1 and 0^0 = 1.

• Unary negatives count. That is, you may use a “−” sign to create a negative number.

• You may use (n!)!, a nested factorial, which is a factorial of a factorial. Nested square roots are also allowed.

• You may use n!!, a double factorial, which is a factorial that uses only the numbers with the same parity (odd or even) as n.

These things are NOT allowed:

• You may not write a computer program to do the puzzle for you — or at least, if you do, PLEASE don’t ruin our fun by telling us all the answers!

• You may not use any exponent unless you create it from the digits 2, 0, 2, 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!)”.

• While we do allow the square root function, you must create any other roots from the digits 2, 0, 2, 0. For example, to take the cube root of a number, use the radical symbol along with (2+0!) to mark it as cube root.

• “0!” is not a digit, so it cannot be used to create a base-10 numeral. You cannot use it with a decimal point, for instance, or put it in the tens digit of a number.

• The decimal point is not an operation that can be applied to other mathematical expressions: “.(2+0!)” does not make sense.

• You may not use the integer, floor, or ceiling functions. You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals.

Helpful Links

• Mathematics Game Worksheet
  For keeping track of which numbers you’ve solved.

• Mathematics Game Manipulatives
  This may help visual or hands-on thinkers.

• Mathematics Game Student Submissions
  For elementary through high school students who wish to share their solutions.

For more tips, check out this comment from the 2008 game. And Heiner Marxen has compiled hints 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.

Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page. And Pat Ballew takes a brief look at the history of such arithmetic puzzles.