Two of the most popular New Year’s Resolutions are to spend more time with family and friends, and to get more exercise. The 2017 Mathematics Game is a prime opportunity to do both at once.

So grab a partner, slip into your workout clothes, and pump up those mental muscles!

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. This year may prove to be a challenge.

## Rules of the Game

**Use the digits in the year 2017 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. You may not use any other numbers.
- Solutions that keep the year digits in 2-0-1-7 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 create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

#### My Special Variations on the Rules

- 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.
- You MAY 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’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

## How To Play

As usual, we will need every trick in the book 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 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 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). My expat daughter is coming home for a visit this month, however, and we’ll be traveling to see extended family. So this tally will almost certainly lag behind the results posted in the comments.

Percent confirmed: 100%

Reported but not confirmed: 0%

Numbers we are still missing: 0%

Wow!

Students in 1st-12th grade may wish to submit their answers to the Math Forum, which will begin publishing student solutions after February 1, 2017. 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, 1, 7 exactly 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 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, 1, 7. 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

- 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.

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.

*2017 Adventures photo by Kitty Terwolbeck and Origami Star by uschi mitzkat via Flickr (CC BY 2.0). New Year’s Resolutions from Wikipedia.*

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I always start with the simplest numbers and operations, then gradually introduce the more potent math tricks. Using only elementary-school math (addition, subtraction, and multiplication), I can do the numbers 1-10. Times-0 is really useful in getting those small numbers!

I can confirm 1-10, and in case we’re interested, I did 2, and 7-10 “in order” (that is, the digits 2, 0, 1, 7 occur in that order in my solution.

I can do many of the others (of the first 10) in order as well, if I start the expression with a unary negative.

Unary negatives are good and very useful!

I’m stumped on how to get 11 with elementary operations, though I did get several of the teen numbers. By adding in decimals, exponents, and factorials, I’ve made it up to 20. And 1-19, I managed to keep in 2-0-1-7 order.

for 11 I did use a factorial. And for 17 I had to use a 2-digit number which I’m not happy about.

Yes, I needed the factorial for 11. For 17, I have the 2-digit solution in my elementary list, but I found a couple of middle-school-level equations, both of which even kept the digits in order.

I got one for 17 without the 2-digit number, but not in order. 🙂

I’m always amazed at how many different ways there are to find a number! Yours must be something I haven’t considered yet. Both of my solutions use the other digits to create a number to which I can add or subtract the seven.

I’m always interested in whether the numbers are in order. I also keep track, for my own challenge, of whether I’m forced to use 2-digit numbers or multifactorials.

Haven’t used either of those yet this year (except for some 2-digit #s in the elementary solutions), but with an odd prime like 7 in our year, I expect I’ll have to at some point. There are only so many things you can do with a seven.

I’ve got 1-78, 80-86, 88-91, 93, 95, 98, and 100. Just missing 7 numbers.

I haven’t tried using double factorials yet, so I’m giving that a try next.

In order, I have: 1-21, 26-28, 31, 32, 34, 37, and 67.

Wow! You’ve been busy. I’ve made it up to 37, but so far I’m stumped on 38.

I try not to use 2-digit numbers, if I can help it, but I needed one to get 29. But I

wasable to get 23-24 and 35 in order.Well, this is small potatoes, but my kiddos (6 and 8) and I did 1-10, plus a few easy numbers after that. I admit to being a bit of a coward and letting them freely mess with the order of the digits. They had a great time, happily.

That’s wonderful! Any time kids “have a great time” playing with numbers, that is certainly not small potatoes. 😀

In my elementary solutions, I don’t bother with keeping the order of digits. I let them go in any order that feels natural and makes sense.

Okay, the double factorials were very useful, especially in the 90s. I’ve now got everything but 79. This is the closest I’ve ever been to a complete list, and missing that one is driving me crazy.

I’ve had to use 2-digit numbers on quite a few, but avoided it whenever I could.

That’s great! I didn’t think it would be possible to solve so many numbers this year.

I’ve made my way up to 50, except I’m still stumped on 38 and 44. I had to use a 2-digit number to get 29, and I needed a double factorial for 46.

Oh, and I did get an expression for 79, too — I even kept the digits in order — so I’m sure you’ll finish your list soon. 🙂

Alright, I just got 79 (with the digits in order, as you suggested), and have completed the game!

My solutions for 38 and 44 are not in order, and both involve getting an expression for a nearby value with 3 of the digits, then adding/subtracting the final digit to get what I wanted. Hopefully that explanation is clear without giving too much away.

Congratulations! I haven’t had time to work on this recently, but I expect to come back to the puzzle when our co-op classes start up in a couple of weeks.

Had car trouble and found myself stuck in town, waiting for help. So I turned back to this puzzle — and surprisingly, managed to finish the list, confirming all numbers.

In past years, I’ve been able to avoid using double factorials, but with this year’s digits I found them indispensable. Repeating decimals and 2-digit numbers came to my rescue more often than I’d like. I always prefer answers that are straight calculations with the single year digits, but it’s satisfying to fill out the list in any way. 🙂

For teachers using the Math Forum: I’m not sure that all the numbers will be possible, since the MF rules disallow repeating decimals. For instance, my solutions for 38, 44, 61, 83, and 87-88 would not be acceptable at the MF website. If anyone finds a solution for one of these numbers

withoutusing a repeating decimal, I’d love to hear about it!