Take the tour ×  There are $16$ natural numbers placed next to
each other. Each is a number from $0$ to $9$. These are in any
order, and you can have as many repeats as you want (e.g. all
$16$ numbers can be zero, or all the numbers can be nine).
Prove that we can always chose consecutive numbers from these 16
numbers such that their product forms a square number. The
conditions are that we must chose at least one number, and the
numbers chosen must be consecutive (i.e. next to each other in
the string of 16 numbers).  I think I should use the Pigeonhole
principle here. Here are a few ideas I had about approaching
this:  There are a total of $10^{16}$ ways of forming a string
of length 16, with numbers from 0-9. These can possibly be the
pigeonholes.  I was thinking the pigeons should be something
like the number of strings which have consecutive numbers such
that their product form a square number.  The largest possible
square number we can have is when all 16 digits are 9, and we
pick all 16 digits (we can