*[Feature photo above by Artis Rams (CC BY 2.0) via flickr. Title background (right) by Dan Moyle (CC BY 2.0) via flickr]*

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza!

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.

## Rules of the Game

**Use the digits in the year 2014 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable by age: Young children can start with looking for 1-10, or 1-25.**

- You must use all four digits. You may not use any other numbers.
- Solutions that keep the year digits in 2-0-1-4 order are preferred, but not required.
- 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, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
- You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
- You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
- You may use a double factorial, but we prefer solutions that avoid them.
*n*!! = the product of all integers from 1 to*n*that have the same parity (odd or even) as*n*.

**[Note to students and teachers:** If you want to take part in the Math Forum Year Game, be warned that they do not allow repeating decimals.]

## How To Play

As usual, we will need every trick in the book to create variety in our numbers. Experiment with decimals, double-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 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 warned: Many teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their homework for them.

*Do not post your solutions. I will delete them.*

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions the last few 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). Today is Kitten’s birthday, however, so I won’t spend much time at my computer. Also, I may be traveling a lot this month, so this tally will probably lag a few days behind the results posted in the comments.

Percent confirmed =96%

1-66, 68-86, 89-92, 94-100.

Reported but not confirmed:

67, 87, 88, 93.

Numbers we are still missing:

None!

Students in 1st-12th grade may wish to submit their answers to the Math Forum, which will begin publishing student solutions after February 1, 2014. Remember, Math Forum 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 may use each of the digits 2, 0, 1, 4 only once in each expression.
- 0! = 1. [See Dr. Math's Why does 0 factorial equal 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. - The double factorial
*n*!! = the product of all integers from 1 to*n*that are equal to*n*mod*2*. If*n*is even, that would be all the even numbers, and if*n*is odd, then use all the odd numbers.

**These things are NOT allowed:**

- You may not use any exponent unless you create it from the digits 2, 0, 1, 4. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(−1)”.
- “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+1)” 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

- 2014 Mathematics Game Worksheet

For keeping track of which numbers you’ve solved.

- 2014 Mathematics Game Manipulatives

This may help visual or hands-on thinkers.

- 2014 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.

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. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.

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Using old Math forum rules, I have the majority of numbers and am missing 53, 57, 67, 85-89 and 91-93.

Hi. Without unary negatives or double factorials, as I has forgotten about those, I’ve got 1-66, 68-85, 89-92, 96, and 98-100.

I’m missing 67, 86-88, 93-95, and 97. So I’m at 92% complete. From Pelf’s comment, it looks like I should focus on 93 and 97 next.

I’m 94% done. I’ve used factorials and double factorials in many solutions. I only used a repeating decimal in my solution for 33, but I’m sure I could find a solution without (my partner did). The numbers I’m missing are: 67, 77, 86, 87, 89 and 91.

I’ve picked up 86, 95, and 97 today.

Still looking for 67, 87, 88, 93, and 94. I guess I’m focusing on 93 and 88 next.

Wow! I’m waaaay behind. I think this puts the confirmed list at 93%.

Reported but not confirmed:86, 88, 89, 91, 93.Numbers we are still missing:67, 87.I’ve only worked a bit in my head, and I couldn’t hold very many numbers at a time. I figured out 1-20, and a few in the 20′s — most of those in 2-0-1-4 order. The rest of the numbers will have to wait until I get caught up on year-end bills and have free time to play on paper…

I get inspired and challenged by those that say they’ve figured out numbers I don’t have. I now have everything except 67 and 87. My solutions for 89 and 86 used a repeating decimal. I’d love to hear if someone found a solution for those without.

I think I’ve found them all! I found a different solution for 89 without using a repeating decimal (based on my solution for 88). I used a repeating decimal in both my solutions for 67 and 87. Without giving too much away, but risking having my comment deleted, part of my solution used sqrt(.(1))=1/3 (square root of point 1 repeating). I *think* that’s correct. Loved to hear if someone got a different answer!

Wow! This may be the earliest we’ve had anyone report 100% solved. :)

I’ve just started, so I’m still focusing on the small numbers and avoiding double-factorials (which I will only allow after exhausting all other options). I’m at 38%, mostly in correct year-order. So far, I’ve had to take the digits out of order for 13, 14, and 33, and I had to use a 2-digit number for 43 and 45.

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