Are your students ready for a challenge?

The Math Forum: 2007 Mathematics Game will be a tricky one:

Use the digits in the year 2007 and the operations +, -, x, Γ·, sqrt (square root), ^ (raise to a power), and ! (factorial), along with grouping symbols, to write expressions for the counting numbers 1 through 100.

- All four digits must be used in the expression.
- Only the digits 2, 0, 0, 7 may be used.
- Multi-digit numbers such as 20, 207, or .02 MAY be used this year.
- The square function may NOT be used.
- The integer function may NOT be used.

Students should be introduced to the fact that 0! by definition is equal to 1. See Dr. Math’s answer to Why does 0 factorial equal 1? for more information.

Dr. Math also tells us that, depending on the context where 0^0 occurs, it can be 1, indeterminate, or undefined/nonexistent. For this game we will accept the value 0^0=1. See the Dr. Math FAQ 0 to the 0 power for more information.

This site is intended for the posting of answers generated by students. Teachers may submit answers for their students. Each answer should include the expression, the name of the student, the student’s grade level, and the name of the school. Use the Web form once for each answer you would like to contribute. If you send more than one solution per form submission, we will be unable to include your answers on our solution page.

Your students may send in their solutions now. Math Forum says they will begin posting student answers after February 1. (I don’t know how long the Web form will be active, once they start posting answers.)

## The Final Total

Percent Solved:76%.

Numbers we could NOT find:

38, 44, 46, 52, 53, 55, 58, 62, 78-80, 82, 86-89, 92-97, 99, 100.

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I played this game in my head last night, while sitting up with sick ds, and I was able to get all the numbers from 1-29 except for 17. (Having 0! defined as 1 is very helpful.) I got a few of the bigger numbers, but couldn’t keep track of which ones without a piece of paper. I really should make a list and be systematic about it, but I doubt I’ll get around to that…

17 is doable. We are in review for final exams now, or else I would give it to my freshmen. Maybe I will make up a packet of optional material over their extra break (state exams that they don’t take, plus 3 more days for who knows what… reorganize for the spring?… they have a 9 day vacation). I could include this, and the fact that the answers are not on the web is a good thing.

Thanks!

17 is more than doable—it’s easy! I don’t know why I couldn’t think of it the other night. Of course, in math, many things are easy only

afteryou solve them.I am getting stuck on 30 and 32

I am still stuck with getting just 16 done and this has to be done by noon today. This isn’t a game for me this is a 50% of my grade!

I don’t know why your teacher would make something like this count for such a big chunk of your grade, but here are some hints that may help you:

Make a list of the digits you have: 2, 0, 0, 7. Add, subtract, multiply, and divide them in every arrangement you can imagine. Remember that 0! is defined as = 1. Add that to your list, because it will give you many more possibilities than 0 alone will give.

Also, you can do other factorials, such as (2 + 0!)! = (2 + 1)! = 3! This will give you more numbers to work with. The more numbers, the more possible ways to combine them.

Although you are not allowed to use squares or other powers whenever you want, you do have a “2” to work with, which will let you square one number. Take advantage of it.

Make a list of all the numbers from 1-100, so that you can write in any ideas for bigger numbers. For instance, you don’t have to work your way up past 30 before doing something easy like 72 + 0 + 0 = 72. If this is for a grade, then be sure to record every possibility you think of.

Best wishes!

Down with a nasty virus this weekend, so there’s not much to do but sit in bed and play math. I

almosthave strength enough to lift a pencil, so I guess I’ll make my list and see what I can come up with. I’m also stumped on 30 and 32 so far, and the closest I’ve gotten to 100 yet is 98…32 is possible. It took me until nearly midnight, and when I saw the solution, it was so obvious that I felt stupid for not getting it days ago. As a math teacher, facts like that should be as easy for me as breathing. (The virus is settling in my chest, so breathing isn’t quite as easy as it used to be. Can I use that as an excuse?)

My score at this point is 60%, and I think I am finished. 30 and 33 remain unsolved, as do 100 and many numbers in between.

OK, got 32.

30 is there, but tricky. I am missing 30, 38, 39, 41,43,44,45,46,52…

I got 100…not easy.

Jonathan

I take it back. I got 700, not 100. Clever, but not the target. I get 90.

Oh, and two comments back, I got 33, not 30. Sloppy typing

Acch, what a mess. I now have 1 – 37. I am missing 38, 43, 45, 46, 53, and then a lot.

Is the any way to share answers? I have 71 solutions and would be happy to trade as I am rather stuck.

K-12 students can submit their answers to Math Forum (see link in article), which will start sharing answers in February. According to the Math Forum page, “For many years mathematicians, scientists, engineers and others interested in mathematics have played ‘year games’ via e-mail and in newsgroups.” So if you are not a K-12 student, but you happen to know one of those e-mail groups, take advantage of it. I suggest you avoid posting answers on the Internet, however, since some teachers are making this a classroom assignment, and there will always be students looking for people to do their work for them.

Denise said: 30 and 33 remain unsolved, as do 100 and many numbers in between.

Like Denise, 32 finally occurred to me in bed last night, and once you realize it, it is so obvious that you feel silly for not getting it earlier! I, too, am missing 30 and 33. 23 took me forever but I finally got it. I am currently up to 37 and have only a few others solved after that (as I’m mostly trying to work through them in order).

This is a fun challenge, and a good thing to give students to fill the empty minutes if they finish something early, since you can do it in little spurts.

Anyone else still working on this? I found a “trick” that helped me solve both 30 and 33. π

I’d love to hear your hint. 30 and 33 are still stumping me.

The rules say that decimals may be used: Multi-digit numbers such as 20, 207, or .02 MAY be used this year.

I found .2 to be particularly helpful in solving 30 and 33…

[slapping forehead] I never even considered decimals. Thank you!

I assume everyone is stuck on 38? I am also missing 43, 45, 46, and fall apart at 52.

Jonathan

jd, I am also stuck on 38. I have 43 and 45, and am missing 44 and 46. I am also missing bunches starting at 52. I do have everything from 63 through 74 inclusive, except 66. I’m pretty happy with my solution for 56, which uses a trick I kept trying in vain to use for 30, and finally found a use for. π

Do we know if the challenge people claim that all the numbers are possible?

My list matches mathmom’s almost exactly. Of the numbers she didn’t specifically mention, I have 54 (but not 56), 60, 76, 81, 90, and 98. My current score stands at 65%, thanks to her reminder about decimals.

The Math Forum people say (on the site linked above): “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.”

I think I goofed when I said I had 44. I have 45. I do have 43 (just now, quite clever, I thought). I have 63 – 76, except 66.

Denise, I think you and I now have exactly the same list of answers, except I have 56. Let me know if you want a hint on that one. jd, I’d take a hint on 75 if you’re willing to share. And 44 if you decide you really do have it. π

Thanks for posting this! I read it during my planning period today, modified the directions to help my kiddos who’d never seen factorial notation and were introduced to exponents on Tuesday, and posted a chart on my brain teaser bulletin board. I had 6 students standing out in the hallway for the whole of their recess period talking about numbers and how to make them. Marvelous!

In my personal list, I’m most proud of 30, because I got it BEFORE seeing mathmom’s hint… And yes, it’s midnight on a Friday night and I’m working on this. No apologies for geekness.

75 is good. Nothing harder than anything else you’ve posted, but it combines them pairwise (using two different tricks), then adds the results.

I have a really nifty 57.

I have 1 – 50 complete except for 38, 44 and 46. After that it gets pretty spotty. Overall, so far I have a 64% – but I’m not done!!

Sara, I think we are pretty close to even. 57 is indeed nifty, but the trick doesn’t extend usefully to any other numbers. I am missing 66, though.

Woohoo! I found a trick that gave me both 66 and 77, as well as alternate expressions for many of the numbers in between.

While searching for the still-elusive 56 and 57, I discovered another trick that gave me 59 and 61. I am now up to 70%, which is far higher than I ever expected a year with two zeros could go.

I got 56! I was hoping something like that would work, and just found it.

I also have a cool expression for 90, (it was dissappointing when it worked out to be 90, and not something cooler and less trivial).

I found 56, too. I also found an illegal expression for 57, and a similar one for 91, but nothing that fits within the rules, yet.

57 and 90 I have, but no progress on 56….

Is there something iffy about the application of the rules?

Jonathan

No. It’s just that things like 14C2 = 91 are not listed in the allowed operations.

And I just got 56, straight up, no cheating!

I still don’t have 57, 66 or 77, but I did just find a trick that gave me 83, 84 and 85!

Oh, I just got 57 also! The trick I used also gave an alternate for 43. Does that sound like the same one everyone else used?

That’s mine, mathmom

im stuck on 44 66 76 87 88 and 89 any answers plzzzzzzzz

I finally found 57. Decimals are just not my thing! I also found 84, but it must not be the same expression as mathmom’s, because I had no “pieces” left over to give me 83 and 85. Oh, and I found a nice expression for 90 (might be the same as Sara’s) to replace the trivial sum I had before.

Current score is 73%. Still missing on my list: 38, 44, 46, 52, 53, 58, 62, and almost everything from 78 up.

I have 72 numbers, probably the same as Denise except missing 55.

I’m particularly pleased with the trick providing my only solutions for 66 and 91 (and of course lots of alternates). It’s almost necessary for 77, but that is susceptible to a similar trick. I’m missing

38

44 46

52 53 55 58 59

61 62

78 79

80 82 83 85 86 87 88 89

92 93 94 95 96 97 99

100

Oops! I forgot to list 55, but I haven’t found it, either. I hadn’t thought to apply the 66 trick to 91, but thanks to Steven, I’ve now added that to my list.

Of the ones on your list, Steven, I know that 59 and 61 are possible, but

verytricky. I am still searching for 83 and 85, because mathmom says she found a way to get them.83 and 85 use a trick similar to the one for 30 and 33, but different. π

Could you give a little hint to get me looking in the right direction for the trick for 55 and 91? It’s still eluding me. Thanks!

Yes, I was just coming to report that I’d found them, when I saw your hint. Double decimals—no wonder it took me so long! As far as I know, there is no trick for 55 yet. For 66, 77, and 91, it turns out that one of the factorials is very close to a certain perfect square, the root of which can be adjusted to give all sorts of numbers, most of which you already have expressions for.

There is at least one other way to get 77, which involves a square root

anda decimal. Tricky!My score now stands at 76%. Does anyone else have any of the other numbers, or do you think it’s possible that we have found everything?

Thanks, that was enough to get me to 66, 77 and 91!

I still don’t have 59 and 61 either, but I’ll keep looking for those.

I’m not totally convinced that we’ve “used up” the trick that worked for 83, 84, 85 — I feel like I’ve just started playing with it, so there may be more it can do.

But I do think we’re coming close to having everything possible, if we don’t already.

I did just get 59 and 61 just now, so I think I have everything everyone else has said they have. I’m almost ready to admit defeat on the rest, but not quite. π

Solutions for the 2007 puzzle are up. I haven’t looked at actual solutions, but just checking which ones there are links for, it looks like we got all of the possible ones. π 2007 Game Solutions

Thanks for posting the link, mathmom! I went to look last week, but they hadn’t put the answers out by then. It’s good to know we didn’t miss anything.

This is brilliant! I found the site revising for the UKMT maths challenge and I’m hooked! In between coursework, I have managed 53 anwsers, but I’m still working on it! Thanks π

Someone needs to post the answer link in the chat conversation because I want to see if there was any other way to get any of the numbers

The answers are at Math Forum: 2007 Puzzle Student Solutions.

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