2011 Mathematics Game

[Photo from Wikipedia.]

Two of the most popular New Year’s Resolutions are to spend more time with family and friends, and to get more exercise. The 2011 Mathematics Game is a chance to do both at once.

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

Here are the rules:

Use the digits in the year 2011 to write mathematical expressions for the counting numbers 1 through 100.

  • All four digits must be used in each expression. You may not use any other numbers except 2, 0, 1, and 1.
  • You may use the arithmetic operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), and ! (factorial). You may also use parentheses, brackets, or other grouping symbols.
  • You may use a decimal point to create numbers such as .1, .02, etc.
  • Multi-digit numbers such as 20 or 102 may be used, but preference is given to 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 factorial of a factorial, which is not the same as a multifactorial
  • n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n
  • n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3

[Note to teachers: The bonus rules are not part of the Math Forum guidelines. They make a significant difference in the number of possible solutions, however, and they should not be too difficult for high school students or advanced middle schoolers.]

How Does It Work?

Use the comments section below to post a running list of the numbers you have been able to calculate. You may also share relatively cryptic tips and hints, but be warned: Some teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their work for them.

Do not post your solutions. I will delete them.

I know of no authoritative list of numbers that can be made with each year’s digits, so we will rely on our collective wisdom to decide when the game is done. We had a lively discussion the last few years. I’m looking forward to the fun!

Keeping Track

As the game results are reported below, I will keep a running tally of confirmed results (that is, numbers reported by two or more players). Today is Kitten’s birthday, however, and we have other busy plans for the weekend, so this tally will lag a few days behind the results posted in the comments.

Percent confirmed = 97%.

Reported but not confirmed =

Numbers we are missing =
76-77, 86.

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

Made it with Math Forum rules = 54%:
1-16, 18-26, 29-33, 36, 40, 42, 45, 49-51, 54-56, 59-61, 64, 66, 70-73, 80-81, 98-100.

Found an expression *without* multi-digit numbers:
1-33, 35-41, 43-55, 57-65, 70-75, 78-85, 87-92, 94-100.

Found a way to keep the digits in order:
1-33, 35-41, 43-55, 57, 58-68, 70-75, 79-81, 83, 88, 90, 92, 94-100.

Math Forum will begin publishing student solutions after February 1, 2011. Remember, you may not submit answers with multifactorials or repeating decimals to the Math Forum site.

Helpful Links

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:

  • By definition: 0! = 1 . [See Dr. Math’s Why does 0 factorial equal 1?]
  • For this game we will accept: {0}^{0} = 1 . [See the Dr. Math FAQ 0 to the 0 power.]
  • Unary negatives are allowed. That is, you may use a “-” sign to create a negative number. This is particularly helpful if you are trying to keep the digits in 2-0-1-1 order.
  • The only digits that can be used to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 1, 1.
  • The multifactorial n !^k = the product of all integers from 1 to n that are equal to n mod k. The double factorial and triple factorial may be written 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 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.
  • No exponent may be used except that which is made from the digits 2, 0, 1, 1.
  • You may not use a square function, but you may use “^2.”
  • You may not use a cube function, but you may use “^(2+0!).”
  • You may not use a reciprocal function, but you may use “^(-0!).”
  • You have to “hit” each number from 1 to 100 exactly — no rounding off or truncating decimals allowed. You may not use the integer function.

With only three distinct digits to work with this year, you will have to use every trick in the book to create variety in your numbers. Especially, you will want to play around with decimals, double-digit numbers, and factorials of all sorts. Remember also that dividing or using a negative exponent creates the reciprocal of a fraction, which can flip the denominator up where it can be more helpful.

For more tips, check out this comment from the 2008 game.

Heiner Marxen has compiled hints, links, 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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32 thoughts on “2011 Mathematics Game

  1. Whew, this is a tough one. I’ve been playing with this on and off all day, and I’m still getting some, so I won’t give a list just yet, but I had a question about the multifactorials. In the rules, you only listed doubles and triples as examples…would higher orders be acceptable? I’ve had some luck with fourth and fifth order multi factorials so far. I’m also just stating that I’m grateful for the vinculum…I think there might only be 30 to 40 possible otherwise.

  2. Only the double and triple factorial are available as “freebies” — for higher multifactorials, you must create an exponent from the year digits. See the section Clarifying the Do’s and Don’ts above.

    I started playing around with this game last night when I couldn’t sleep, and managed to do 1-26 plus a few higher ones (99 is easy) in my head. I had forgotten about the vinculum, but I did use a couple of double factorials.

    And I came up with a fun (though illegal) expression for 48. Since 24 is relatively easy to get with the Math Forum rules, I could use the 22-factorial to double it:
    [24]!!!!! !!!!! !!!!! !!!!! !!
    And then I take the 46-factorial of that, to get 96:
    [48]!!!!! !!!!! !!!!! !!!!! !!!!! !!!!! !!!!! !!!!! !!!!! !😀

  3. Heh, good point made with the multifactorials. Okay then, I’m sitting at 75/100 right now…and of those, a full 41 were found using the vinculum, so that “bonus rule” is definitely having an impact on my list. At this point, I have:

    1-33, 35-37, 39, 40, 42, 44-46, 49-57, 59-67, 70, 71, 75, 79-83, 88, 90, 92, 97-100

    Full disclosure, I did pull out my list at a family dinner last night and got a few good ideas from there. I had been almost ignoring ordinary decimals until that point, and slapped myself over a few obvious ones, as well as finding simpler methods for a few trickier ones I already had. Definitely good fun.

  4. I’ve gotten 1-31, 33, 36, 37, 40, 42, 45-51, 53-55, 59-61, 63, 64, 66, 70, 72, 79-82, 90, 96-100. Making free use of multifactorials (2d and 3d order) and the vinculum. I’ve been trying to keep the digits in order wherever possible — of the above, only 42, 51, 82, and 97-100 required taking them out of order, and only a few of the rest required me to use double-digit numbers. To do it I had to use some nested multifactorials, but the whole thing amused me more when I could read the year left to right in the solution.

    I may get back to this and look for more, but that’s where I got stuck for today.

  5. I can confirm many of the above, and I’ll add 38, 94, and 95.

    Of the numbers reported as solved, I’m still missing 39, 52, 57, 62, 65, 67, and 75.

    Also, I haven’t figured out how to keep the digits in order for 64, 79, or 81, but I’ll be looking for David’s way…

    I got 48 of the numbers with Math Forum rules. The rest have required the vinculum, multifactorials, or both. I don’t think I used any nested multifactorials, though. My brain can’t handle numbers that big!

  6. Okay, I’ve picked up several that David reported that I had missed, but it looks like Denise already confirmed those. I also managed to get 41, 78, and 91, which appear to have been unreported thus far. Of those three, only 41 has the digits in order. So I’m at 82/100 right now.

    I have NOT been able to get 38, 94, or 95 yet, but I will keep working at it.

  7. I haven’t had much time to play with this, thanks to a nasty flu bug that hit my family. Poor Kitten had her worst birthday ever!

    Still, I can confirm 41. I’ll keep working on the others.

    I double-checked my calculations for 38, 94, and 95. They’re definitely not Math Forum numbers. I don’t know how you got 35 or 97, but on my list all five of these numbers are related.

  8. I can confirm 38, 39, and 57. Also I’ve gotten 58. I must not have gotten 38 the way Denise did; in my solutions 38 and 58 are closely related, and definitely not to 99. None of those four are following Math Forum rules, but 39 and 57 keep the digits in order.

  9. Using the math-forum rules I found: 16, 73, 80 and 81. Except for 73 they all require multi-digit numbers. And 73 also keeps the digits in order.

  10. My 7th graders are having a great time with this activity. We spend the first 15 minutes of class (ok, maybe it’s a little more than 15) adding to the class list. We have class vs class competition going (5 classes). I can confirm 81, 16, 64, and 73 using Math Forum rules.
    Do you think summation would be allowed?

  11. No, sorry. Summation is not allowed.

    I can confirm 62 and 91. (My kids helped me think through 91 while we were waiting in the doctor’s office. Nasty flu!) I think that brings us up to 81% confirmed solutions. Wow!

    I also found the Math Forum rules numbers Corrie and Isabel reported, and I felt like slapping my forehead over a couple of them. Playing around with the new toy (multifactorials) distracted me from looking at the more basic options. BTW, 81 is possible using Math Forum rules without a multidigit number.

    I still can’t get 64 with Math Forum rules or the digits in order, and I’m still missing 52, 58, 65, 67.

    And I’m also waiting for someone (anyone?) to confirm 94 and 95.

    But if the puzzle was too easy, it wouldn’t be fun, right?

  12. A few more additions:
    I found 41 and 58 (not following Math Forum (MF) rules, digits in the right order, no multi-digit numbers).
    I found 64 using the MF rules.
    I did 72 and 98 without multi-digit numbers
    and 75, 83, 98 and 100 with digits in the right order.

  13. Well, this evening I found 73, which I think was already confirmed, and I can now confirm 94, but not 95…I don’t know if I found it the same way you did or not. I also managed to get 74, which doesn’t seem to have been reported yet.

    So, I’m still missing 38, 58, and 95 from those reported. I thought I had a brilliant solution for 58…then realized I was taking a factorial of a fraction…so that didn’t work out. I think I’ve made that same mistake in past years😦

    Let’s see…65 was related to how I got 64 and I also have 64 with the digits in order, so if you solve the one, you’ll probably solve the other. Also, 67 was a tweak of how I got 70. Hopefully that helps.

  14. And more…
    I can confirm 52,94 and 95.
    I found 34, 41, 43, 84 and 89.
    You can add 43 and 52 to the list of numbers that you can make with digits in the right order and 43, 53, 84 and 89 to the ‘without multi-digit numbers’-list.

  15. I can confirm 58 and 65, which by my count brings us up to 86% solved. And surprisingly, the Math Forum rules are up to 53%.

    As fast as I find things, though, you guys keep adding to my list. Of the confirmed numbers, I am still searching for 52. I’m also looking for a single-digits solution for 98 and for a way to do 83, 98, and 100 with the digits in order.

    And I’m missing all of the “reported but not confirmed” numbers: 34, 43, 67, 74, 84, 89.

  16. BTW, my solutions for 94 and 95 were closely related, simply a matter of shifting the brackets.

    I think I tried that factorial-of-a-fraction trick once. My most common mistake this year has been using 20111.😦

  17. I can confirm 34, 74, 84, 89. I finally found 52, AND I can add two new numbers: 85 & 87.

    Still looking for 43 & 67, plus the digits-in-order version of 83.

  18. I can confirm 67 with digits in the right order (using a 3-digit number) and 85.
    Still struggling with 87…
    For 83 with digits in the right order I have 2 + 81; 43 = -5+48 (if that helps).

  19. I believe Math Form rules permit nested factorials and possible double factorial (!!), so that may increase the MF rules count.

  20. Wow, I set this down for a couple days, and when I come back you’ve almost solved the entire thing! Let’s see, I found a lot of the confirmed numbers that I was missing. I think the down time let me see some of the numbers in a new light.

    I can also confirm 68 and 69, Corrie’s hint made them a lot more obvious than they otherwise might have been.

    I’m still looking for 34, 85, 89, and 93. Since we’re so close, I’m also gonna take a closer look at 76, 77, and 86, since that seems to be all that’s left for a perfect game. But I’m sure everyone will be doing that.🙂

  21. The blog post updates are running behind, but I can confirm we have 97% solved — and 54% using Math Forum rules. Amazing!

    And Nth_X said, the only numbers left are 76, 77, and 86. I haven’t had time to look into those any further. It seems a shame to get so close to a perfect game and not be able to finish it. Any ideas?

  22. By comparison: Last year we got 49% with Math Forum rules. When we added the vinculum bonus (to use repeating decimals), that brought our 2010 total up to 74% complete.

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