Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
Explore a number pattern which has the same symmetries in different bases.
We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.
Is there a general pattern of $1$'s and $0$'s in a binary number
which is the square of a binary number containing only $1$'s? Try
squares of $11$, $111$, $1111$ etc? To generalize you might use the
fact that the sum of powers of two is a geometric series.