Programming handles repetitive calculations so you can focus on finding patterns in the results. Using programs written on a graphing calculator, you generate two new sequences from the natural numbers by flipping or reversing binary digits. Analyzing at least 1,000 terms reveals surprising patterns — some obvious, some hidden — that you then explain with mathematical formulas.

Finding repeated shapes, numbers, or features in data is what makes it possible to sort or identify things — and binary sequences turn out to be a rich place to look. When you convert counting numbers to binary and reverse the order of digits before converting back to decimal, the resulting sequence reveals surprising patterns. Some are obvious at first glance and some are hidden. Deriving formulas to explain them across at least 1,000 terms confirms that the repetitions are real and mathematically predictable.

When you convert counting numbers to binary and then flip each digit — zeros become ones, ones become zeros — the resulting sequence follows a predictable subtraction rule from computer math. Reversing the order of binary digits before converting back to decimal reveals a second sequence with its own patterns, some obvious and some hidden. You write programs on a graphing calculator to generate both sequences across at least 1,000 terms, then derive formulas to explain what you find.