Let’s resolve to have fun with math this year. Ben has posted a preview of 2018’s mathematical holidays. Iva offers plenty of cool ways to think about the number 2018. And Patrick proposes a new mathematical conjecture.
But my favorite way to celebrate any new year is by playing the Year Game. It’s a prime opportunity for players of all ages to fulfill the two most popular New Year’s Resolutions: spending more time with family and friends, and getting more exercise.
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 2018 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 2-0-1-8 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 family has been going through the annual ritual of pass-the-flu-bug, however — and whenever we finish that game, 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: none.
Numbers we are still missing: none.
Wow! If you are still working your way through the puzzle (as I am), take heart — it can be 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, 2018. 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, 8 exactly once in each expression.
- For this game, 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.
- 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, 8. 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.
2018 Equation photo by Iva Sallay and 2018 Sparkles by NordWood Themes on Unsplash. Semiprime Blues cartoon by Ben Orlin.
Denise Gaskins' Let's Play Math 
I haven’t done the New Year math game for at least a couple of years. I tried challenging myself more than usual. I’ve gotten most numbers, the ones I’m still missing are: 31, 33, 43, 47, 51, 53, 76, 93, 94 and 98.
I did the following using single digits in order: 1-16, 20, 24, 27, 32, 35, 40, 42, 45, 48, 52, 54, 58, 68.
The following are in order, but use double digit numbers: 17-19, 21, 23, 25, 28, 29, 38, 90.
I used a vinculum in the following: 34, 37, 67, 69-73.
I haven’t used any double factorials (yet) but may be breaking them out to figure out solutions to the outstanding numbers.
My most interesting math fact I discovered: .2^(-0!)=5
I also love that dividing by .(1) is the same a multiplying by 9.
As in the past, I’ve become obsessed with this, I think that’s why I haven’t done it for the last couple of years, it occupies my mind for far too much time.
I looked back at past year’s comments that I had a lot of input on finding solutions. The hints there helped me recall some techniques I used in the past, but only helped me get a couple more numbers. I’ve come to the conclusion that there is more flexibility with smaller digits, the 8 is just so limiting in what can be done.
Looking back at past years, there were typically only 1 or 2 numbers that can’t be figured out, and they were usually rather large-ish. I can’t for the life of me get 31. I mean, 31!!!! It’s such a small number, how hard can it be? Seems like it’s pretty hard.
So here is what I have left unfound: 31, 33, 43, 47, 51, 93, 94 and 98.
92% completion, I should feel happy with that, but somehow it’s unsatisfying. Waiting for other’s comments so I can find inspiration/tips to tackle some more.
I haven’t had much time to play, but I can confirm the numbers 1-21. I got them all in 2-0-1-8 order, but I had to use a two-digit number for 19.
I’m pretty happy with my progress this year. I’ve found expressions for everything except 93 and 98.
I’ve used the vinculum for a good number of those, including 31, which was mentioned as a difficulty.
I got to 96% and was stuck, then I broke out the !! for two more. Now I’ve just got two to go!
Okay, I’ve now made it as far as 30 (except for 26), keeping the digits in order. I also found an expression for 33.
I did have to use 2-digit numbers for 19, 23, 28 and 29.
I got inspired and tackled some more. I also only have 93 and 98 left to get. I felt I was getting close to 93, but I have to go somewhere.
My initial solution for 31 used a double factorial, but then I found a second one using a vinculum. Looks like 47 might be the only one that I used a double factorial for. For the most part, double factorial has proven unhelpful. Also, my calculator does not have double factorial, if a fancy graphing calculator doesn’t include the function, I feel like I’m cheating using it, so I’m trying to avoid it.
I’m curious about Denise’s solution for 33. Did you use an “interesting” method for multiplying by 3 using only a 1? Well, that’s what I did, curious if you found an alternate method.
I’m quite happy with my 98% completion now. I’ll keep checking in to see if anyone reports getting 93 and/or 98.
No, I had forgotten that “interesting” method. I’ll have to give it a try on some of the numbers I have left. My trick for 33 was much simpler: 25 + 8. Of course, you first have to figure out how to get the 25…
Also, I can think of a way to get 47 without a double-factorial, if you take the digits out of order. So I think you’re right, Johanna: the “cheat” isn’t worth much this year.
I got 48% keeping the digits in order. But now I’ve switched to using them in any order, and gotten my percentage up to 63%. Still have quite a ways to go to catch up to you both. 🙂
I figured out 93 using a double factorial. I also figured out an alternative to 47 without using double factorials. My solution for 48+/-1 would also get me 49. Strangely I didn’t use that for 49, my solution for 49 is oddly complicated, using 2 different repeating decimals.
As I type this, in the back of my mind I’m thinking “how does one get 25 with 2,0 and 1? Oh! I see it!”. So much you can do with decimals and negative exponents. ;-P
Done! 100% yay!!!! Even found an alternate answer to 93 without double factorials. Decimals to negative exponents proved invaluable (and in the case of 98, square roots really helped).
Phew! I don’t know if I ever got 100% previously. Whoo-hoo!
do you have to use all the numbers
Yes, Xander, you have to use all four numbers: 2, 0, 1, and 8. And if possible, try to keep them in that order — but you can’t always do that. Whatever the order, all four digits must be used.
Our family also got all of the numbers (as a family effort)… 🙂 lots of factorials used, some repeating decimals.
Impressive! I’m inspired to know there are solutions to all the numbers. Now I just need to find time to discover them…
This is really hard! My friend and Ive been trying so hard because our teacher gave it to us as a project! we only have 23 more problems and we want to know if anyone could help us with 90?