*photo by Creativity103 via flickr*

For our homeschool, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the ** 2012 Mathematics Game**!

## Rules of the Game

Use the digits in the year 2012 to write mathematical expressions for the counting numbers 1 through 100.

- You must use all four digits. 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, .01, etc.
- You may create multi-digit numbers such as 10 or 202, but we prefer solutions that avoid them.

Bonus Rules

You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.You may use multifactorials:

n!! = a double factorial = the product of all integers from 1 tonthat have the same parity (odd or even) asn.n!!! = a triple factorial = the product of all integers from 1 tonthat are equal tonmod 3

[Note to teachers:Math Forum modified their rules to allow double factorials, but as far as I know, they do not allow repeating decimals or triple factorials.]

## How To Play

With only three distinct digits to work with this year, we will need every trick in the book to create variety in our numbers. Experiment with decimals, double-digit numbers, and factorials of all sorts. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it may 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, 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 found by two or more players). Today is Kitten’s birthday, however, so I won’t spend much time at my computer. Also, I’ll be traveling a lot this month, so this tally will lag a few days behind the results posted in the comments.

Percent confirmed =97%.

Reported but not confirmed =

77, 92.

Numbers we are still missing =93.

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

Middle school rules =68%.

Old Math Forum rules, no repeating decimals or multifactorials:

1-32, 34-44, 48-52, 58-65, 70, 72, 74, 80, 90, 94-95, 97-100.

New Math Forum rules, confirmed =77%.

NOT Math Forum:

33, 55-57, 66-67, 69, 71, 73, 77-79, 81-84, 86-89, 91-92.

Needed multi-digit numbers:

44, 67-68.

Could NOT keep the digits in order:

29, 31, 33, 37, 39, 41, 44, 55, 59, 65, 67, 69, 71, 73, 76-78, 89, 91, 95.

Math Forum will begin publishing student solutions after February 1, 2012. Remember, you may not submit answers with triple (or higher) factorials or repeating decimals to the Math Forum site.

## 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:**

- . [See Dr. Math’s Why does 0 factorial equal 1?]
- The only digits that you can use to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 1, 2.
- Unary negatives count. That is, you may use a “-” sign to create a negative number.
- You may use (
*n*!)!, a nested factorial — a factorial of a factorial. Nested square roots are also allowed. - The multifactorial = the product of all integers from 1 to
*n*that are equal to*n*mod*k*. You may write the double factorial and triple factorial as !! and !!!, respectively, but for higher multifactorials BOTH*n*and*k*must be constructed from the year digits.

**These things are NOT allowed:**

- “0!” is not a digit, so it cannot be 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.
- You may not use any exponent unless you create it from the digits 2, 0, 1, 2. 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)”.
- You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals. You may not use the integer, floor, or ceiling functions.

## Helpful Links

- 2012 Mathematics Game Worksheet

For keeping track of which numbers you’ve solved.

- 2012 Mathematics Game Manipulatives

This may help visual or hands-on thinkers.

- 2012 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.

Fun Stuff! I am going to send over a few participants….

Great! The more, the merrier.

Kitten has spent the morning playing with her slumber party friends, so I’ve had time to do a little math puzzling. I think the double factorial option will be very handy for Math Forum students — and since it’s based on odd and even numbers, it should be easy to explain.

1, 2, 3, 4, 5, 6, 8, 9, 10

I can confirm John’s numbers.

I’ve been trying to think about this as my middle-school math club students will, and I don’t think they’ll have too much trouble with the numbers from 1-24, except 15. I’ve got all those under the old Math Forum rules, but 15 needs the new rule.

(I also got them all with single digits and in order, but I don’t think my students will be comfortable enough with decimals and powers to do that.)

10, 25, 48, 52, and 100, all using a way to get 50.

Hi, Climbing Gecko! It’s good to “see” you. 🙂

I can confirm your numbers (including 50 — did you get that, too?), though I didn’t get all of them the same way. In fact, the only one of those I got by starting with 50 was 52, but it’s fun to collect several different ways to do the numbers.

I can think of two ways to get 50 with a leftover 2. One of them keeps the digits in order.

More from my quest for numbers my middle school students may be able to find:

25-27, 34-37, 39, 41, 44, 97-100.

Hi All. I’ve been using this with some of my high school classes to get them thinking “mathematically” again after the break…so I’ve had a lot of time to look at this. So far, I’ve got (or can confirm):

1-52, 60, 63-65, 72, 79-81, 83, 89-91, 95, 98-100 (68/100)

I’ve got several students who have really had their imagination captured with this game, and are pushing me to check and verify new numbers several times a day 🙂

I can confirm some of the above, including: 1 – 26, 29 – 33, 35 – 41, 43, 45, 47, 49 – 51 and 63 – 65.

Wow, nth_x, your classes have been busy! My co-op class doesn’t meet until mid-January, but I hope they enjoy the puzzle as much as yours.

I can confirm all of nth_x and Sara’s numbers, and I’ll add the following:

59, 61-62, 70, 74, 94.

Here’s some more from me and/or my students:

53-55, 58, 62, 66-69, 74-76, 82, 96-97

Ah, you’ve passed me up now. I can confirm some of those, but I’m still missing 53 and 66-69.

Still, I can add a few you didn’t list: 56, 71, 85.

I can confirm 61 of the updated list of missings/unconfirmed.

I couldn’t sleep last night, so I lay on my pillow staring at the ceiling and solving numbers. I can confirm 53, 66, and 68-69.

I’m pretty sure I thought of a way to get 67, too, but I can’t remember it this morning. 😦

And I found two new numbers: 57, 78.

Edited to add:Aha! I remember 67.I can confirm 56. I can also get 26 and 35 with Math Forum rules and the digits in order.

Thank you, David!

I can confirm 59, 70, 71, 78, and 94. I also have a solution for 73 (single digits, not in order, using double factorial and repeating decimal).

Hi, GT! I can confirm 73, which brings us up to an amazing 91% confirmed. Wow!

I also figured out how to get 61 and 63 in order.

this is just way too hard and torture! & i have to do this for math credits!

ive only gotten 1-14 , 17-25, 32, 39-42, 51, 60, 98, 100 ..

Hi Yeargame Puzzlers,

I administer the yeargame over at the Math Forum these days, and I wanted to write to say I *will* be accepting solutions with multifactorials this year. I was educated about them in early January and have updated the rules to include them. I love that mathematicians see a good idea and extend it as far as possible…

If someone could help me understand how repeating decimals might be used, I’ll see if we can accept those answers as well.

Thanks for playing!

Max

Hi, Max! Thank you for stopping by. The repeating decimal is most useful as a way to access bigger numbers than are otherwise possible.

For example, something ÷ .(1) — using brackets to indicate the repeating 1, since I can’t type a vinculum — gives me the equivalent of multiplying by 9. Handy!

Hmm… I’m a bit skeptical about repeating decimals. It does unlock some new numbers and is a great teachable moment. But “repeat” isn’t really an operation, or if it is it’s on the level of Int, Trunc, Rnd, etc. And it’s sort of like introducing “divide by 9” as a fair operation which is an odd one to include. But I could still be convinced.

7 years from now, in 2019, I definitely would stay away from repeating decimals since we have a 9 in there already.

I don’t think using repeating decimals is any more artificial than allowing a decimal point or multi-digit numbers. Neither of those are operations, either. And according to Wikipedia, some people disallow the square root symbol (in the Four 4’s Puzzle) because it adds an implied “2”.

I prefer the use of repeating decimals over multfactorials, because I think repeating decimals are solidly within the standard prealgebra curriculum topics — but it’s a matter of taste, like preference in ice cream. Multifactorials feel to me like an artificial trick. Not that that has kept me from using them, but I avoid them when possible.

Back from out of town, I’m ready to update the game count.

New numbers that need to be confirmed:77, 84, 86-88, 92.

Also still waiting for confirmation on 57 and 85.

There is only ONE number I haven’t been able to make this year: 93.

I added a new category:“Middle School Rules” are the same as the old Math Forum rules, without repeating decimals or multifactorials. My students are still struggling with regular decimals, and normal factorials are a new idea for them — I don’t want to confuse them further. Still, I’ve found

~~63~~64 numbers that they can calculate, if they are persistent.In the spirit of Denise I wrote a Prolog program that allows repeating decimals but not multifactorials. It found an expression of depth 8 for 57. Of interest (to Chinese students) it solved 88 (maintaining the order: 2 0 1 2). It found solutions for all numbers from 0 to 100 except for 67 68 69 77 92 93.

Hi, Lew!

I’m not sure what “an expression of depth 8” means, but I agree that 57 is a toughie. Your program found expressions for quite a few of the numbers I needed multifactorials for. That gives me some new puzzles to shoot for — finding the nonmultifactorial versions.

67 and 69 are possible without multifactorials, too.

I’m coming to the game late, but I have 28-31 solved with the old MS rules (without needing multi-digit numbers).

Thanks, MathMom! I guess I need to get back to it. My math class did find a solution for 30, which I forgot to note above, but I’ll have to play around with 28, 29, and 31…

From where should i play this game?

You play the game in your own mind and on scratch paper. The challenge is to make as many of the numbers as you can figure out. You can ask questions in the comment section here, and see which numbers other people have made, and report any new numbers that you find. But the game itself is solitaire — just you and the math.