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.
Problem setup
The beauty of this solution is that you do not need to solve for the solutions
110 in any order.
Plans to Solve/Investigate the Problem
The plan for finding solutions involves starting with the given example and
beginning to add and take away subtractions and additions of 3 and 4. Later we
will change the multiplication of 3 * 4 to 4 * 4 when we begin exhaust problem
solutions.
Investigation/Exploration of the Problem
Start with the given
example, 3 x 4  3  3 = 6, but eliminate one subtraction of 3:
3 x 4  3 = 9
Change the subtraction of
3 to a subtraction of 4, and you have:
3 x 4  4 = 8
There seems to be no
reason why you cant use the example:
3 x 4  3  3 = 6
Change one subtraction of
3 to a subtraction of 4:
3 x 4  3  4 = 5
Change both subtractions
of 3 to subtractions of 4:
3 x 4  4  4 = 4
Now add another
subtraction of 3:
3 x 4  4  4 3 = 1
It would seem that we have
exhausted solutions for problems starting with 3 * 4 = 12, so now we shall try 4
* 4 = 16 and subtracting. (3 * 3 = 9 likely will not give us a large enough
number to leave room for sufficient subtractions). We now have problems for the
solutions 9,8,6,5,4, and 1, leaving problems to be written for 2,3,7 and 10.
Lets try to find problems resulting in 2 or 3 figuring we can make a slight
change of a 3 or a 4 to get the other solution.
We discover a solution for
10 accidentally:
4 * 4 3 3 = 10
Adding another subtraction
of 3 gives us:
4 * 4 3 3 3 = 7
Adding a subtraction of 4
to that problem gives us:
4 * 4 3 3 3 4 =
3
And finally, changing one
subtraction of 3 to a subtraction of 4 givens us the final solution:
4 * 4 3 3 4 4 =
2
We found the solutions 1 to 10 using only these five keys: 3, 4, x, , = in the
following order: 9,8,6,5,4,1,10,7,3,2. The order of solutions was arbitrary; we
were concerned with generating new problems. This is an important idea for
students attempting to solve the more challenging extension.
Extensions of the Problem
A
good extension of the problem would be to give students the task of finding all
solutions from 1 to 25, and to use the numbers 1,2,3,and 4, but in any order.
Each problem should contain each number used only once, and students may use any
operation including division. Solutions to the problem would include use of
grouping symbols, and this lesson would be an excellent addition to a unit on
order of operations.
Author &
Contact
Pat Devane
Memorial Middle
School
Conyers, GA
pjdevane@rockdale.k12.ga.us
