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### Twenty or Fewer Steps

```Date: 08/28/2002 at 10:52:20
From: Joan Wellener
Subject: Pre-algebra. problem

Problem:  Using only the number 1 and addition, subtraction,
multiplication, and division and the equals sign, find the answer in
20 or fewer steps. The answer is 75.
```

```
Date: 08/28/2002 at 13:19:21
From: Doctor Peterson
Subject: Re: Pre-algebra. problem

Hi, Joan.

A 12-year-old has asked essentially the same question, but with some
extra information that makes it a bit clearer:

Only the 1, +, -, x, /, (, ), and = keys on a scientific calculator
are working. How can a result of 75 be reached by pushing these
keys fewer than 20 times?

This specifies that you are using a calculator, and are allowed to use
the parentheses. Even then, calculators differ enough that I'm not
sure my answer will work. Here it is: on many calculators, you don't
need to keep pressing the keys for the second number and the operator
if you want to repeat an operation, but you can just press "="
repeatedly. For example,

1 + = = = =

gives the answer 5, because each "=" adds 1 (the number in the display
when the first "=" is pressed) to the current number in the display:

1 + (1) = 2(+1) = 3(+1) = 4(+1) = 5

Now you can press "*=" (times, equals) to multiply this by itself:

1 + = = = = * =

results in 25.

Now we have to multiply this by 3, and for that we need the
parentheses, or something else like a memory key on a simple
calculator. We can continue with

* ( 1 + 1 + 1 ) =

to get 75. My calculator, at least (the one on a Windows computer)
can't repeat addition of 1 within the parentheses, so we can't save
the extra "+1"s. But we've used a total of 17 keypresses to

Does that meet your needs, or do you have different requirements?

Here is another answer from two years ago:

=====================================================================
Here's a hint: start with 111 - 11, which gets you to 100 after six
keystrokes (of course you need an operator key (+,-,/,=).  If
you need more help to finish it, please write back.

Including the final equals sign (=), there is a solution in 19
keystrokes.

- Doctor Douglas, The Math Forum
=====================================================================

This implies that there is a more conventional solution. I looked on
the Web to see if this puzzle is discussed anywhere, and found a page
where a programmer discusses how he wrote a program to find all
possible solutions and see if he could get under 19 keystrokes. His

Calculating Coffee - January 2000 - David Detlefs
http://www.unixreview.com/documents/s=1471/urm0000ja/

"Here is the problem: you have a simple electronic calculator
that's just a little bit broken - of the digit keys, only the "1"
works. Can you get the calculator display to show the number
"75" in fewer than 20 keystrokes? The calculator has the four
basic arithmetic functions: +, -, x, and /; an = key, and left
and right parenthesis keys. Unlike some calculators, it is not
the case that n* = displays the square of n."

He found that there are 21 solutions, all exactly 19 strokes, of
which he shows only one. It's not at all tricky; he does a series of
subtractions, evaluates the result with an equals, multiplies that
and does some more subtractions. Apparently all it takes is a little

After realizing that, I just took Dr. Douglas's start and found a
solution that uses nothing but subtraction!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculators, Computers
High School Puzzles
Middle School Puzzles

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