Practice Invariants in Math

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

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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.

Showing a random 20 of 50 problems.

Example 1

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Three numbers 1,1,11,1,1 start on a board. A move replaces (a,b,c)(a,b,c) with (a,b,c+1)(a,b,c+1) for some chosen coordinate. Is the parity of a+b+ca+b+c invariant?

Example 2

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You translate a triangle 5 units to the right. Which property is invariant: area, position, or both?

Example 3

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You change every $1\$1 bill in a stack to four quarters. Is the total dollar amount invariant?

Example 4

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On a 3×33\times 3 grid you place +1+1 or −1-1 in each cell. You may flip all signs in a row or column. Is the product of all 99 entries invariant?

Example 5

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Stretching a rubber band changes its length. Is length an invariant under stretching?

Example 6

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When you rotate a square, what stays the same — its area or its position?

Example 7

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On a board, you may replace two numbers a,ba,b with a+ba+b. Start with 1,2,3,4,51,2,3,4,5. What is invariant, and what is the final number?

Example 8

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A sequence starts at 1 and each term is 3 times the previous minus 2: an+1=3an−2a_{n+1} = 3a_n - 2. Show that the quantity an−1a_n - 1 grows by a factor of 3 each step (i.e., an−1=3n−1(a1−1)a_n - 1 = 3^{n-1}(a_1 - 1) is an invariant structure).

Example 9

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Three jars hold 3,5,83,5,8 liters. A pour doubles one jar by transferring from another. Is the total 1616 invariant, and can a jar reach 00?

Example 10

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Is the perimeter of a polygon invariant under rotation?

Example 11

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A bag has 77 red and 33 blue marbles. You swap two marbles' positions. What is invariant — the count of red marbles?

Example 12

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A frog jumps on a number line by ±3\pm 3 each move, starting at 00. Is its position mod 33 invariant?

Example 13

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You rearrange 3+53+5 into 5+35+3. What is the invariant?

Example 14

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Folding a piece of paper in half — is the paper's area invariant?

Example 15

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On a clock, you advance the hour hand by 12. Is the displayed hour invariant?

Example 16

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A chess knight moves on an infinite board. Show its color (of the square it stands on) alternates each move, so 'color after nn moves' has parity invariant in nn.

Example 17

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Two different objects both weigh 55 kg. Does sharing the invariant 'weight =5=5 kg' make them identical?

Example 18

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Numbers 1,2,…,20241,2,\dots,2024 are on a board. You erase two, aa and bb, and write a+b−1a+b-1. After many moves one number remains. Find it.

Example 19

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Reflecting a triangle across a line — is its area invariant?

Example 20

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On a 4×44\times 4 checkerboard with 88 black and 88 white squares, can 1515 dominoes (each covering 11B+1+1W) tile any 1515-square subset?
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Related Concepts

EqualAlgebraic Invariance