Problem of the Week #3
(Spring 2014)
A natural number n bigger than 10 is called a “super-square” if any number formed by
two consecutive digits in n (without changing their order) is always a perfect square. For
example, 8164 is a super-square since the numbers 81, 16, and 64 are perfect squares. Other
examples of super-squares are 25 and 649.
How many super-squares are there?
Solution:
Notice that the only perfect squares of two digits are
16,
25,
36,
49,
64,
81.
(1)
Any number formed with the first two digits of a super-square must belong to the list (1).
We thus count the number of super-squares for each case.
• The number 16 is a super-square. If we wish to include a new digit after 16 to form
a new super-square, that digit must be 4 since 64 is the only number in list (1) that
starts with 6. Thus, we see that the only super-square with three digits that starts
with 16 is 164.
Since 49 is the only number in list (1) that starts with 4, then the digit 9 is the only
one that can be attached to 164 to form a new super-square.
Now, no number in list (1) starts with 9, so it is not possible to create a new supersquare by attaching a new digit to 1649. We thus obtain that the list of super-squares
that start with 16 is
16, 164, 1649.
• Since there are no numbers in list (1) that start with 5, we have that the only supersquare that starts with 25 is
25.
• Following the same rationale used above to find the super-squares that started with
16, we see that the super-squares that start with 36 are
36,
364,
3649.
• Similar to the case of 25, the only super-square that has 49 in its first two digits is
itself
49.
• The only super-squares starting with 64 are
64,
649.
• Finally, the super-squares that start with 81 are
81,
816,
8164,
81649.
Counting the super-squares obtained above in each case, we conclude that there are 14
super-squares.