Practice Invariants in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Quantities or properties that remain unchanged during a process, operation, or transformation—values that stay the same no matter how the system is rearranged or acted upon.
Rearranging an equation keeps both sides equal—equality is the invariant.
Example 1
mediumA sequence starts at 1 and each term is 3 times the previous minus 2: \(a_{n+1} = 3a_n - 2\). Show that the quantity \(a_n - 1\) grows by a factor of 3 each step (i.e., \(a_n - 1 = 3^{n-1}(a_1 - 1)\) is an invariant structure).
Example 2
hardIn a game, you can add 3 or subtract 5 from a number. Starting at 0, can you reach 1? Use an invariant (parity or modular) argument.
Example 3
mediumShow that for any right triangle with legs \(a, b\) and hypotenuse \(c\), the quantity \(a^2 + b^2 - c^2 = 0\) is an invariant (Pythagorean theorem).
Example 4
hardA 2×n grid of squares is colored with 2 colors. You repeatedly flip all colors in any row or column. Show that the parity of the number of black squares is an invariant.