Title:
Finding Magic Numbers
Problem Statement: Given the "magic triangle" shown,
place the digit 1 through 9 in the circles in a very
special way, using each digit once and only once. For
each side of the triangle, the sum of the two "inside
circles" subtracted from the sum of the two "end
circles" must be the same "magic number." Find the
magic triangles whose magic numbers are 0, 3, and 9.
Problem Setup: The objective of this investigation is
to find missing numbers. I must keep in mind that
there are certain numbers that are already in place for
example, the "inside circle", must be subtracted from
the sum of the two "end circles" and the two end
numbers must be the same magic number. There must be a
pattern of numbers that are used to add and subtract
evenly to solve this investigation.
Plans to Solve/Investigate the Problem: In solving
this investigation, I plan to brainstorm several
numbers that I can evenly compute to give me my magic
numbers of 0, 3, and 9. The first thing I would do is
create a triangle.
I would start with one circle as
the point and from that point going down, connect lines
to each circle there after. Your diagram should
illustrate the following, 1 point circle, three circles
on each side of the point circle, going down adjoined
with lines and two circles on the base of the triangle,
adjoined with lines - completing a full three sided
triangle. I would arrange the numbers in the circles
on the triangle to support my statement above. The
following numbers will be arranged on the triangle
inside of the circles. I would put an 8 as the point
of the triangle. The following numbers on the left
side of the triangle from top to bottom would be 4, 3
and 2. The numbers on the base of the triangle
starting from the two are 5, 6 and 9. The numbers on
the right side of the triangle going from bottom to top
starting from the 9 are 1, and
7. To verify that the numbers in the triangle are
accurate based on my objective, I would use simple
addition and subtraction to find the magic numbers.
I would start with the number 8 and add it to the 9,
which is 17. I would add the 7 to the 1 and that will
give me 8. I would then subtract the 17 from the
8 - that would give me 9, which is a magic number. I
would repeat the same steps on the left side of the
triangle. 8 plus 2 is 10. I would take the inside
numbers and add them up; which are 3 and 4 to equal 7.
I would subtract the
10 and the 7, that would give me my second magic
number, which is 3. To find the last magic number, I
would repeat the same steps in finding the other two
magic numbers. I would add both the outside numbers
(2+9 = 11) and then add the inside numbers (5+6=11),
take the two sums and subtract them (11-11=0), that
will give you the last magic number - 0.
You may
display your triangle using Microsoft excel.
Extensions of the Problem: You can extend this problem
by challenging the students to create a pattern of
numbers using the magic numbers of 0, 3, 9. Have the
students describe the pattern, then write the next
three numbers.
Author & Contact: Travis L. Brown contact at
TLopezBrown@Aol.com