Title
Combinations

Problem Statement
On a calculator you are allowed to use only these five keys: 3, 4, x, -, =. You can press them as often as you like. Find a sequence of key presses that produces a given number in the display. For example, 3 x 4 - 3 - 3 = will produce 6. Find a way to produce each of the numbers from 1 to 10. Clear your calculator before each new sequence.

(Source: Adapted from Mathematics Teaching in the Middle School, Nov-Dec 1997)

Investigation/Exploration of the Problem

I started by opening a spreadsheet so that I could see everything easily Click here for spreadsheet. I started with the obvious i.e. 4-3 = 1 (see chart below).

I then wanted  to find what combination would yield 3. I know that 4x3-(3x3) = 3 BUT the problem states that we can't use parentheses so I need to find another way to write 3 x 3 using the given symbols ( 3, 4, x, -, =).

3 x 3 is the same as 3 + 3 + 3. Knowing this,  we can re-write the combination as 4 x 3-3-3-3 = 3. Using this information it was easy to find the other combinations by starting at 4 x 3 and subtracting from there. For example, to find the combination that yields 2, we can start with 4 x 3 = 12. We know that 12-10=2 so we need to find what combination of 4 and 3 equals 10. The only combination of 4 and 3 that equals 10 is 4 + 3 +3; thus yielding the combination 4*3-4-3-3=2. The pattern holds until we reach 7. Here we have to start at 4*4 or 16 and subtract from there because 4*3 or 12 is too small and will not yield the desired results.

 (4 - 3) = 1 4*3-4-3-3= 2 4*3-3-3-3= 3 4*3-4-4= 4 4*3-3-4= 5 4*3-3-3= 6 4*4-3-3-3= 7 4*4-4-4= 8 3*3= 9 4*4-3-3= 10

These are not the only combinations that will work. For example,  4*3 - 3 = 9 and 4*4 - 4 -3 = 9 and there are others. The key is to try and think of combinations that yield the desired result and go from there! Good Luck!

Author & Contact
Laura Thomas

Memorial Middle School

Conyers, GA
lthomas@rockdale.k12.ga.us