*feature photo above by Alan Klim via flickr*

New Year’s DayNow 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

For many homeschoolers, 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 ** 2013 Mathematics Game**!

## Rules of the Game

Use the digits in the year 2013 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 203, 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 small numbers 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, 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 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 =100%!

[The number 83~~is under review~~has been confirmed.]

Reported but not confirmed =83.

Numbers we are still missing =none.

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

The following data are incomplete, pending further investigation:

Middle school rules =92%.

1–42, 44–75, 77-81, 84, 89–91, 92–100.

Old Math Forum rules, no repeating decimals or multifactorials.

New Math Forum rules =98%.

All the above, plus 43, 76, 82, 85-87.

NOT Math Forum:

83, 88.

Needed multi-digit numbers:none?

[I’m still relying on multi-digits for several numbers, but Bill in the comments says he found ways around them.]

Could NOT keep the digits in order:

34, 37-38, 41, 45, 49, 51-52, 54-55, 59, 61, 65, 68, 76, 79, 81-84, 86-91, 93-95, 97-99.

Math Forum will begin publishing student solutions after February 1, 2013. Remember, your students 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:**

- 0! = 1. [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, 3.
- 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
*n*!^{k}= 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, 3. 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

- 2013 Mathematics Game Worksheet

For keeping track of which numbers you’ve solved.

- 2013 Mathematics Game Manipulatives

This may help visual or hands-on thinkers.

- 2013 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.

Printed a copy out for my daughter and I to work on. My partner wanted his own after a day. I have 94 completed, I’m missing 77, 83, 86, 87, 88 and 99. My daughter has 41 completed, and has had a great review of exponents, order of operations, and learned about factorials. It’s been a great learning experience for her during the holidays!

Wow! I’ve been traveling and haven’t even had time to start, yet, except for trying to figure out a few numbers mentally. I usually do this at night when I’m supposed to be falling asleep — it beats counting sheep, but I have a hard time remembering the next morning which numbers I got.

I’ve got 77, 83, 86, and 99. So if Johanna and I are right that leaves only 87 and 88?!

Shortly after I posted my comment, I found a website where someone had written a computer program to come up with solutions. It was rather old, but I looked at the solutions for 2003, and I found solutions for 77 and 99, which used similar techniques I had been trying for those answers, but just hadn’t quite gotten there yet. In my search for a solution for 87, I “accidentally” got 77, so that only leaves 99 as a cheat. So the only ones I haven’t solved now are 83 and 88. I’m inspired now knowing that 83 is at least possible (assuming David used the “new rules” method). Sounds like 88 is the “holy grail” of 2013 (maybe?)

I just found 88! Unfortunately it’s using a repeating decimal, so I don’t know how that fits in with the rules here. Unless someone can figure out how to turn a 3 into an 8, it’s the only solution I can find… (still working on 83)

Repeating decimals are allowed here, but not at Math Forum. I still haven’t gotten started on this, since we aren’t starting school until next week. Looks like I’ve got a lot of catching up to do!

BTW, if you run out of numbers to do for this year, I had two numbers from last year’s game that were never confirmed (77 and 92 — and 93 that nobody ever found). Would you like to try finding them?

Well, the holidays are over, so I have to get back to work, thus ends this fun activity for me (my partner gave up at about 80% solved). It’s interesting the patterns you find in math, for example, did you know that sqrt(7!+43^2) = 83? Crazy! And OMG, my mother-in-law is coming tonight, don’t distract me with more challenges (must… resist… the urge… to do more math!!!!) ;-P

😀

Okay, I finally got started. As is my habit, I’m playing by the old Math Forum rules first (no multifactorials or repeating decimals). So far, using those rules, I managed to find 1-40, though I had to use a double-digit number for 39, and I wasn’t able to get 33, 34, 36, or 37 in year-number order. I still have a lot of catching up to do….

I found everything, except for 88 (like most people). I’ll try to give an overview with the other information.

With old math forum rules I found 1 – 42, 44 – 75, 77, 79 – 81, 84, 89 – 91, 93 – 100.

I needed double factorials (new math forum rules) for 43, 86, 92.

Numbers I found not math forum rules (repeating decimals or higher level factorials): 76, 78, 82, 83, 85, 87

Keeping the digits in order worked for 1 – 16, 18 – 27, 29, 30, 32, 33, 35, 36, 40, 43, 44, 47, 48, 53, 56, 64, 66, 67, 69 – 75, 77, 85, 92, 96.

I used a multi-digit number for 13 and 84. I can do 13 without a multi-digit number, but then I don’t have the digits in order.

I got two expressions for 13 (and also for 17 and 61): one with double digits, and another one with the numbers in year-order.

I’ve worked my way up to 75, but I’m still missing several along the way. I haven’t really begun to work at the higher numbers yet, although I did hit 78, 80, and 100 while I was trying to figure out other things.

It is possible to do 78 with the old Math Forum rules and the numbers in order.

I have all but 77,85,87,89, and ‘holy grail’ 88.

Hello. This is my first time here. Nice place you’ve got!

This is the first year I’ve done the Year Game; I’ve also introduced it to my students. Some of them are almost as obsessed as I’ve become!

Using current Math Forum Rules, I have every number except for 83 and 88. Using quadruple factorials, I can get both of those. For the record, I’m ONLY using single digits, although I’m allowing my students to use multi-digit numbers.

However, I have NOT even attempted to keep all the digits in order. That said, I think I know how to get 78 using digits in order using the New Math Forum rules.

I’ve come up with some curious expressions, but interestingly, I have NOT used square root.

Square roots were wonderful on the years when one of the digits was a perfect square. Still, I’ve used square roots this year for 20, 71, and 73 so far, to keep the digits in order.

I’m still way behind you guys. I’ve made it up to 77% complete. Still missing: 43, 52, 54-56, 67, 76-77, 82-88, 92-99.

Yes! 88 confirmed. Not with the digits in order, not using the (old or new) math forum rules, but it is possible.

Ok, this is now officially driving me nuts. I was the first to get 88, but I *still* haven’t gotten 83!!!! I was going to offer to trade hints for 88 for hints for 83, but now I’m left with just asking: Can someone give me a hint on 83? Bill Nut mentioned using quadruple factorials, but that didn’t help me find an answer. I didn’t use anything more than a double factorial in any of my other solutions, and it sounds like others have found it without using higher order factorials, but I may be wrong. Anyone feel like sharing?

Hi, Johanna,

Hmmmm – I don’t know what the procedure is. I’d be delighted to share a hint or two about 83 if you share a hint about 88!

How about this: I used three of the four digits to reach 81, and then used a quadruple factorial on the final digit to reach 83.

Does that help?

Oh, I think I should have read your earlier comment more thoroughly. To make quadruple factorials, you need to create the power as well as the base. I limited our game to triple factorials.

So did anyone get 83 or 88 without using factorials higher than triples?

I haven’t given up on my old Math Forum rules, yet. I’m up to 86% without using multifactorials or repeating decimals, and still hoping to find a few more…

I apparently misunderstood what a quadruple factorial was. I was thinking originally that it was n!!!!, but I found it defined as (2n)!/n! on-line. I didn’t even know that double factorials (!!) existed before taking this challenge on, so I’m learning all kinds of new things here!

I have a hard time understanding how a single digit can be used in either notation to get anything useful out of it. Using (2n)!/n! requires the use of at least 2 digits, unless there is some other way of denoting this, which I don’t know, or can’t find on-line. I found a combinatorics notation for it, but I thought that wasn’t allowed, and still requires the use of 2 digits. If there is a link to further explanation, I’d appreciate it.

I’m already exploring trying to find 83+/- n with only 3 digits as a possible solution. No dice so far, but I also haven’t devoted a large amount of time to explore it since the weekend. I’m going to spend a large quantity of time in a hockey rink this weekend again, so I may take paper, pencil and a calculator with me again. There’s something about arena air that stimulates the mathematical mind!

As for 88, once I found an answer, it seemed so easy. I’ve already hinted that there is a repeating decimal (and the confirmed answerer made the same statement, so I suspect she got the same answer). I also “technically” only needed 3 digits in my answer. That’s all I want to reveal for now.

Anyone else have a different possible solution for 83 that they want to share? Or, a resource to learn more about quadruple factorials would be interesting as well.

Hi Denise, You were posting a reply as I was composing mine! I haven’t done a comprehensive breakdown of my solutions, but I thought I’d just mention that all my solutions comply to the new math forum rules, although the digits aren’t always in order, and I do use multi-digit numbers. The only exception is 88, where I used a repeating decimal.

I’m curious now if 83 is possible without higher order factorials, or is it the “holy grail” of 2013?

If 83 is the “holy grail”, at least it’s an attainable one. Try a repeating decimal + a unary negative + a square root, all single digits, but not in order.

Does that mean you’ve gotten in Denise? Argh! I’m getting increasingly frustrated!

Phoooooo, cleansing yoga breath….

OK, I’m ready to have another go at it….

In case anyone is wondering, Denise just gave the answer away to 83. I sat down, and wrote the answer on the first line of my sheet of paper. It was *that* easy.

100% complete, whoo-hoo!

Sorry, clarification. 98% completed by me, thanks to Denise for the solution to 83, and Heiner Marxen (see link to his site above) for the solution to 99.

So how does it work? Do we post our solutions here after February 1? Or is there another site where the teachers can post their solutions? (Forgive the newbie!)

Please don’t post your answers here. You can share your percentage complete, or the numbers you’ve found, but not how to get them (except for relatively cryptic hints). Teachers assign this puzzle for extra credit throughout the year, so there will be students looking to find easy answers even throughout the summer.

I got distracted working on the math carnival, so I haven’t made much progress, I added 43, 52, and 55 under the new math forum rules, which brings my percentage up to 90%. But some of the numbers I’ve found must have other solutions, according to your statistics, because I’ve used double-digits or other tricks where you reported not needing them — so I’m still looking…

OK – but IS there a place where I can share my findings – and more importantly, where I can find out how people got that elusive 83 and 88? Is there a way of exchanging info privately? (I’m not shy – I’ll post my e-mail!)

Seems only fair to share another hint about 88, since I’ve gotten solutions for all the numbers (Denise posted how to do 83 above). I’ve hinted previously that I used a repeating decimal. I searching to figure out how someone could come up with a repeating decimal solution, I had to think about how one gets a repeating decimal. The answer: division. (bonus hint, use 88 as the divisor). The answer almost literally jumped out of my calculator.

Wow, Johanna, now I’m even more confused! I think I’ll stick with trying to find the easier numbers for now.

Bill, I don’t know of any place that’s set up for adults to share answers. You can email me if you want, or you can wait and see whether any students submit answers for those numbers to the Math Forum site.

I will tell you that the solution for 83 just uses basic facts. When you finally get the answer, it’ll probably be a “How could I miss that?!” moment.

Trying to explain how the thought processes work inside my mind isn’t easy. I don’t think it’s easy for anyone. In thinking about how to more clearly explain how I got to my solution, it suddenly popped into my head that 88 can be solved using very similar basic facts as used for 83. In fact, using 2 digits for calculating 81, the remaining 2 digits can be used to find everything from 81+/-7. One technique, 15 answers. Math is amazing!

Ah, and before I leave it at that, I made a mistake in my “hint” about 88. The 88 should be the dividend. So, by trial and error, I just started dividing 88 by what ever other numbers I could find using a single digit only, and as I said before, an answer jumped at me from my calculator (in the form of a repeating decimal using the remaining digits).

“One technique, 15 answers. Math is amazing!”Ah, of course! That brings me a lot closer to done. I’d still like to find Math Forum expressions for these as well, and I still need a few of the 90’s…

Responding to Johanna (that sqrt(7!+43^2) = 83), I have my own list of such coincidences:

7=sqrt(6!!+1)

11=sqrt(5!+1)

16=sqrt(10!!/5!!)

21=sqrt(9!!-7!*.1)

27=sqrt(6!+9)

31=sqrt(6!!*20+1)

62=sqrt(10!!+4)

64=sqrt(7!-9!!+1)

71=sqrt(7!+1)

720=sqrt(10!/7)

I tend to play by Math Forum rules plus “a radical b” (because when you already have a sqrt symbol, who’s to say how you use it). This allows 88.

Hi!

For a twelve-grader, this is my sixth year of yeargaming. I followed the old math forum rules mostly and got 1-81 (except 43), 84, and 89-100. (76 seemed to be attainable with a power function and single factorials?) With new rules, I have 43, 85, 86, but am still missing / confused over 82, 83, 87, 88.