20070807, 16:18  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Student error.
A student wrote out this equation 123456789 = 100. Can you insert plus and minus signs in between digits of the number on LHS. to make it right ? You could use even two digits together. Its not necessary to only take single digits and the plus and minus signs need not be equal in number. Try it at the most for 7 signs. Can you do it with only 3 plus or minus signs ? Mally Last fiddled with by mfgoode on 20070807 at 16:23 
20070807, 18:08  #2 
May 2004
New York City
2·2,099 Posts 
1234567+89 = 100 What if we allow the other operations and parentheses? For example, 1+2+3+4+5+6+7+8*9 = 100. Are there other such representations? 
20070807, 18:20  #3 
Jun 2003
17×281 Posts 
Code:
123+4567+89=100
123+45+6789=100
1234567+89=100
1234567+89=100
12+3+4+567+89=100
12+34+5+67+8+9=100
1234+56+7+89=100
1+234+56+7+8+9=100
1+234+5+6+789=100
1+2+345+678+9=100
1+2+34+5+6+78+9=100
Last fiddled with by axn on 20070807 at 18:30 
20070808, 11:39  #4  
Bronze Medalist
Jan 2004
Mumbai,India
100000000100_{2} Posts 
Other operators.
Quote:
In this problem I only gave the simplest of cases and the problem can be complicated enough, by using fractions and allowing zero as the tenth digit,and not allowing other operations like multiplications. The example you have given does not qualify for using *. But you may lump two digits together and maintain the rule to only the use of a max. of 7 signs. HINT: make use of the minus sign too. You are correct in the 3 sign rule. Excellent work! If you can get the 7 sign one then these two solutions ( 7signs and 3 signs) are the only few of the very few solutions possible for the 9 digits. I have got only two of the 7 sign ones at hand. There may be more in this computer era as these problems were based on precomputer days. We will then extend the problem to fractions and ten digits. Best of luck! Mally Last fiddled with by mfgoode on 20070808 at 11:42 

20070808, 12:12  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
The works !
Quote:
One observation with strained eye I have made. You have also given Davar's three sign one. In the good old days this was considered the only combination we could get. It will be worthwhile to find another. I mean 123  45 67 + 89 = 100. Best of luck! Mally 

20070808, 13:22  #6 
Jun 2007
Moscow,Russia
7·19 Posts 
Some other ( may be all solutions), which include multiplication and division operations also, described at http://hi.baidu.com/%C9%BD%C9%BD%BA%...61d9583b0.html
Last fiddled with by VolMike on 20070808 at 13:24 
20070809, 17:27  #7  
Jun 2003
12A9_{16} Posts 
Quote:
So, to answer your question, what I had posted earlier was the exhaustive list of solutions for 100 (again, the case of a leading minus sign is not accounted for). So there is only one solution with three signs. 

20070810, 06:08  #8 
Aug 2002
Ann Arbor, MI
433 Posts 
I suppose it wouldn't be too much more difficult using a bruteforce method, but what about solutions using addsubtractmultiplydivide where you evaluate like in RPN [ie, 1+2345*6789 would be (((1+23)45)*768)9]?

20070810, 08:35  #9  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Pascal program.
Quote:
Thank you axn1 once again. I am not into programming at all and perhaps it will be a diversion to me, and too late a stage in my life, to take it up now. However people like you, who have mastered both ends of the spectrum of modern Maths, are what is required today. If you have at least one sign + in your solutions along with the minus terms, can't you put it first so that there is no need specifying this in your programme as a minus to be leading ? I'm sure this can be done. So Old Kordemsky who wrote "The Moscow Puzzles' back in the old days was right after all that the 3 sign solution was, and hearing from you probably, the only solution. It will be great if you coud step up your search for another 3 sign solution It will make history! Mally 

20070810, 08:44  #10  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Gallop ahead!
Quote:
Thank you Kevin for extending the problem to ' divide multipy' solutions. We can do that but lets finish off with plus and minus solutions as I have got a few related problems dealing in two signs Plus and minus, which we can tackle. This is a result of skimming thru all the puzzles books I have and these are considerable in number ( I dont even know how many). And Im presenting only number theory puzzles for our members as a collection of them for ready reference. I will be back to present some more problems Mally 

20070810, 16:53  #11 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Fractions.
Well I am back again. 1) Using the digits 0 to 9, only once each, find fractions with plus or minus signs to total to 100. Eg: 1 3/6 + 98 27/54 + 0 = 100 2)What is the unique whole number whose square and cube between them use up each of the digits from 0 to 9 ? 3) Compose two fractions using digits from 0 to 9 whose sum shall be equal to unity ? 4) Given 6 different digits forming a number beginning with 28. By transfering 28 to exteme rght hand side we get the new number thus formed which is exactly double the original number Find the number. Mally 
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