Practice Invariance in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

A property of a mathematical object that remains unchanged when the object undergoes a particular transformation or operation.

What stays the same when things change? That's often the key.

Example 1

easy
Show that the sum of the digits of a multiple of 9 is always a multiple of 9. Verify with n = 198 and n = 729.

Example 2

medium
A sequence starts at 1 and each step either doubles the value or adds 3. Show that the parity (odd/even) of the value changes predictably and identify an invariant.

Example 3

easy
Show that the expression x^2 + y^2 is invariant under the transformation (x, y) \mapsto (-x, -y).

Example 4

medium
In a game, you start with the number 6. Each move you may subtract 1 or divide by 2 (if even). Show that the quantity n \pmod{3} is not always preserved and find an invariant that is preserved.
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