# Fun with digits

Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100.

We got one answer

12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

and suggested there existed at least one more. I do not claim to have made an exhaustive search but there appear to be more than just two answers. One of this is

123 + 4 - 5 + 67 - 89 = 100

I am sure there it at least another one. Want to find it?

There is a keen observation that in the two examples above at least one of the operations is subtraction. And this is also true for all *additive* (the ones in which the only allowed operations are addition and subtraction) examples below. In fact, it is impossible to avoid subtraction even if the digits come in an arbitrary order. To figure out why, it may be helpful to recollect the notion of digital roots.

You may allow for operations other than addition and subtraction. This leads to a completely new set of problems with numbers having fractional parts. Variations include setting targets other than 100. Here, for example, a representation of one that uses all ten digits:

1 = 148/296 + 35/70

There are many ways to merrily spend time solving arithmetic problems. One way is to attempt representing numbers with limited means. For example, I can represent 100 with five threes as

In the 1960s another kind of number puzzles have become very popular. Cryptarithms are brain teasers obtained when digits in numerical calculations have been replaced by letters. Customarily, distinct letters stand for different digits. Stars substitute for any digit and are not related to each other.

I received the following letter from Belgium:

From: Gui et Nicole RULMONT

Date: Tue, 22 Apr 1997 17:02:44 +0200

Dear Cut-the-Knot,

First please excuse my English. I am Belgian and I am very interested by your site!

You wrote in "Fun with digits": Start with the sequence of non-zero digits 123456789. The problem is to place plus or minus signs between them so that the result of thus described arithmetic operation will be 100.

Some years ago, il found in the french magazine *Science et Vie* the 11 solutions:

1 + 2 + 34 - 5 + 67 - 8 + 9 = 100

12 + 3 - 4 + 5 + 67 + 8 + 9 = 100

123 - 4 - 5 - 6 - 7 + 8 - 9 = 100

123 + 4 - 5 + 67 - 89 = 100

123 + 45 - 67 + 8 - 9 = 100

123 - 45 - 67 + 89 = 100

12 - 3 - 4 + 5 - 6 + 7 + 89 = 100

12 + 3 + 4 + 5 - 6 - 7 + 89 = 100

1 + 23 - 4 + 5 + 6 + 78 - 9 = 100

1 + 23 - 4 + 56 + 7 + 8 + 9 = 100

1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100

If we put a "-" before 1, we have one more solution:

-1 + 2-3 + 4 + 5 + 6 + 78 + 9 = 100

Using the "." decimal separation I found another solution:

1 + 2.3 - 4 + 5 + 6.7 + 89 = 100 (solution of my own)

What about 987654321 ? There are 15 solutions, said *Science et Vie*:

98 - 76 + 54 + 3 + 21 = 100

9 - 8 + 76 + 54 - 32 + 1 = 100

98 + 7 + 6 - 5 - 4 - 3 + 2 - 1 = 100

98 - 7 - 6 - 5 - 4 + 3 + 21 = 100

9 - 8 + 76 - 5 + 4 + 3 + 21 = 100

98 - 7 + 6 + 5 + 4 - 3 - 2 - 1 = 100

98 + 7 - 6 + 5 - 4 + 3 - 2 - 1 = 100

98 + 7 - 6 + 5 - 4 - 3 + 2 + 1 = 100

98 - 7 + 6 + 5 - 4 + 3 - 2 + 1 = 100

98 - 7 + 6 - 5 + 4 + 3 + 2 - 1 = 100

98 + 7 - 6 - 5 + 4 + 3 - 2 + 1 = 100

98 - 7 - 6 + 5 + 4 + 3 + 2 + 1 = 100

9 + 8 + 76 + 5 + 4 - 3 + 2 - 1 = 100

9 + 8 + 76 + 5 - 4 + 3 + 2 + 1 = 100

9 - 8 + 7 + 65 - 4 + 32 - 1 = 100

Write the sign "-", three solutions:

-9 + 8 + 76 + 5-4 + 3 + 21 = 100

-9 + 8 + 7 + 65 - 4 + 32 + 1 = 100

-9-8 + 76 - 5 + 43 + 2 + 1 = 100

With the decimal point:

9 + 87.6 + 5.4 - 3 + 2 - 1 = 100 (solution of my own)

If I "shuffle" the digits there are many solutions. I found some when I was young, for example:

91 + 7.68 + 5.32 - 4 = 100

98.3 + 6.4 - 5.7 + 2 - 1 = 100

538 + 7 - 429 - 13 = 100

(8×9.125) + 37 - 6 - 4 = 100 etc etc etc ....

very interested by cryptarithms and I collect them. Do you want to receive french cryptarithms ? Do you know non-english cryptarithms ? Thanks!

Anthony Lesar notes that the solution

**Note**: There is a whole bunch of pages that offer practice problems of this kind. Also, Inder Jeet Taneja has gathered a fantastic collection of various sequential representations of numbers from 1 to 11111.

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