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All the King's Digits

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When I was a boy in school, I always wanted to "make a 100" on my exams. I'm sure you also want to do the same. Unfortunately, I didn't always make that 100, because it's difficult to do sometimes.

Here is a way, however, that you and I can "make a 100", and it's easier to do. You see, using all the ten digits, we will make numerical expressions that have the value of 100.

A simple example of this is the following:

100 = 2(1 + 9) + 8(3 + 7) + (4 + 5 + 6)(0)

Here are the main rules:

  1. All ten digits (0, 1, 2, 3,..., 9) must be used once and only once to form the expression.
  2. All the basic operations symbols are permitted: addition (+), subtraction (-), multiplication (x, or *, or use of parentheses), and division (/ or ).
  3. Other functions allowed are: square root (sqrt), raising to a power (^), and factorial (!).
  4. You may use decimal points.
  5. You may place two or more digits side-by-side to form larger numbers. (This is called juxtaposition.)
  6. You may use parentheses ( ), or square brackets [ ], even nested together in the same expression, if it helps you to correctly form 100.

Two other useful techniques are permitted in this activity. The first of these is called "summation". This normally employs the Greek letter "sigma" (which can't be made in e-mail), but its shape somewhat resembles a capital letter "E". Therefore, to demonstrate its use in this activity, "summation(4)", or "E4", will mean "the sum of the numbers from 1 to 4". Hence, E4 = 10, because 1 + 2 + 3 + 4 = 10.

The other item is the "cube root" function. Cubes and cube roots are very important concepts in basic and higher math due to their use in volume problems. So we will allow two ways to do this. First, you may use fractional exponents like this: 8^(1/3). That expression has the value of 2. But we will also permit the use of "cbrt" to mean the cube root of the number. This releases the digits 1 and 3 to be used elsewhere in your expression, if you desire. An example could be: cbrt(27) = 3.

Finally, one very important rule remains. The presence of the zero (0) must be essential to the evaluation of the expression. This means, if it were removed or deleted from your expression and the result still is 100, then such expression is not valid. Notice, that in the example given above, if the 0 were taken away, the value of the expression becomes 115. So, the 0 served a definite purpose in the expression.

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