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 
accomplish the task.

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 
version adds an important restriction:

   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 
perseverance to find an answer.

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/