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Problem of the Week
Problem B and Solution
Reflect on These Numbers
Problem
Suppose that all ten single digits are written in the font shown below:
a) Which four digits are symmetric about a horizontal line through the middle?
b) What three-digit numbers are symmetric about a horizontal line through the
middle? You may use each digit more than once in total, but only once in
each three-digit number.
c) Of the three-digit numbers found in part b), which are also symmetric about
a vertical line through the middle? Explain your thinking.
Solution
a) The only numbers in the given set with a horizontal line of symmetry are
0, l , 3, 8 .
b) Any three-digit combination of the numbers in part a) that does not start
with 0 will have a horizontal line of symmetry.
Those beginning with l are l03, l08, l30, l38, l80, and l83.
Those beginning with 3 are 30l, 308, 3l0, 3l8, 380, and 38l.
Those beginning with 8 are 80l, 803, 8l0, 8l3, 830, and 83l.
c) There are three digits with both a vertical and a horizontal line of
symmetry, 0, l, 8 . But for a three-digit number to have both
horizontal and vertical symmetry, it would have to be a palindrome such as
808, l0l, or 888. Since we can use each digit only once in each three-digit
number, there are no such numbers in our set.
Something to Think About
How would your answer to part b) change if you could use the digits more than
once in any three-digit number?