Invariants Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Invariants.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An invariant is a quantity that holds steady no matter how you transform or rearrange the system.

Common stuck point: The procedure for invariants is the easy part; the trap is assuming any unchanged-looking quantity is the invariant. Asking "Is there a property that holds equal before and after the transformation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a property that holds equal before and after the transformation?

Worked Examples

Example 1

medium

A 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).

Answer

a_n = 1

for all

n

; fixed point is an invariant

First step

Define

b_n = a_n - 1

. Then

b_{n+1} = a_{n+1} - 1 = (3a_n - 2) - 1 = 3a_n - 3 = 3(a_n-1) = 3b_n

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Example 2

hard

In 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

medium

The sum of digits of a number doesn't change modulo 9 when you add 9. Verify: 47 → 47+9=56. Is the digit sum invariant mod 9?

Example 4

medium

Numbers

1,2,\dots,10

are on a board. You repeatedly erase two and write their sum mod

11

. What is invariant, and what is the last number?

Example 5

medium

Numbers

1,2,\dots,n

are on a board. You may replace any two

a,b

with

|a-b|

. Show the parity of the sum is invariant.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Show 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 2

hard

2 \times n

grid of squares is colored with

2

colors, where

n

is a positive even integer. You repeatedly flip all colors in any row or column. Show that the parity of the number of black squares is an invariant.

Example 3

easy

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

Example 4

easy

You rearrange

3+5

into

5+3

. What is the invariant?

Example 5

easy

Solving an equation, you add

4

to both sides. What stays invariant?

Example 6

easy

A bag has

7

red and

3

blue marbles. You swap two marbles' positions. What is invariant — the count of red marbles?

Example 7

easy

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

Example 8

easy

Scaling a triangle by factor

2

— is its number of sides invariant?

Example 9

easy

Two different objects both weigh

5

kg. Does sharing the invariant 'weight

=5

kg' make them identical?

Example 10

easy

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

Example 11

medium

On a board, you may replace two numbers

a,b

with

a+b

. Start with

1,2,3,4,5

. What is invariant, and what is the final number?

Example 12

medium

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

Example 13

medium

You add the same number to every entry of the list

\{4,9,4\}

. Is the RANGE (max minus min) invariant?

Example 14

medium

Numbers

1

10

are written. You repeatedly erase two and write their difference. Is the parity of the total sum invariant?

Example 15

medium

A frog jumps on a number line by

\pm 3

each move, starting at

0

. Is its position mod

3

invariant?

Example 16

medium

4\times 4

grid is colored like a checkerboard. You place dominoes covering two adjacent squares. What invariant does each domino preserve?

Example 17

medium

You repeatedly multiply or divide a positive number by

2

. Is whether it is a power of

2

times the original invariant?

Example 18

medium

Is area invariant under rotation but not under scaling? Justify with a unit square.

Example 19

medium

Coins show

5

heads,

3

tails. A move flips any two coins at once. Is the parity of the number of heads invariant?

Example 20

challenge

Numbers

1,2,\dots,2024

are on a board. You erase two,

a

and

b

, and write

a+b-1

. After many moves one number remains. Find it.

Example 21

challenge

5\times 5

board has a piece on each square. Each move shifts every piece to an adjacent square. Show pieces on black squares stay invariant in count parity. Why can't all pieces leave?

Example 22

challenge

Three jars hold

3,5,8

liters. A pour doubles one jar by transferring from another. Is the total

16

invariant, and can a jar reach

0

Example 23

easy

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

Example 24

easy

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

Example 25

easy

When you multiply every entry in a list by 2, is the median invariant?

Example 26

easy

You change every

\$1

bill in a stack to four quarters. Is the total dollar amount invariant?

Example 27

medium

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

n

moves' has parity invariant in

n

Example 28

medium

A grasshopper at

0

jumps

\pm 7

each move. Can it reach

50

Example 29

medium

On a

4\times 4

checkerboard with

8

black and

8

white squares, can

15

dominoes (each covering

1

+1

W) tile any

15

-square subset?

Example 30

medium

Numbers

a,b,c

on a blackboard transform as

(a,b,c) \to (b+c, a+c, a+b)

. Is

a+b+c

invariant?

Example 31

medium

Three numbers

1,1,1

start on a board. A move replaces

(a,b,c)

with

(a,b,c+1)

for some chosen coordinate. Is the parity of

a+b+c

invariant?

Example 32

medium

A point on a

5\times 5

grid moves

\pm 1

horizontally or vertically each step. Is its grid-color (checkerboard) invariant?

Example 33

medium

On a number line, you may add

2

or subtract

3

. Starting at

0

, can you reach

1

Example 34

medium

8\times 8

board has its two opposite corners removed. Can the resulting

62

squares be tiled by

31

dominoes?

Example 35

hard

Numbers

1,2,3

are on a board. A move replaces them with

\frac{a+b}{\sqrt 2}, \frac{a-b}{\sqrt 2}, c

(for chosen

a,b

from the three). Show

a^2 + b^2 + c^2

is invariant.

Example 36

hard

On a

3\times 3

grid you place

+1

-1

in each cell. You may flip all signs in a row or column. Is the product of all

9

entries invariant?

Example 37

hard

n\times n

chessboard,

n

even, is to be tiled by

L

-shaped trominoes (3 squares). For which

n

is the count of squares (

n^2

) divisible by 3?

Example 38

hard

Two stacks have

a

and

b

tokens. A move adds

1

to one stack and subtracts

1

from the other. Is

a + b

invariant?

Example 39

hard

The digit sum of a positive integer

n

satisfies

S(n) \equiv n \pmod 9

. Use this to find the digit sum of

9^{100}

modulo 9.

Example 40

hard

4\times 4

grid is filled with

\pm 1

. You may flip all entries in any single row OR any single column. Can you always make all entries equal to

1

Example 41

challenge

Numbers

1, 2, \ldots, 100

are on a board. A move erases two numbers

a, b

and writes

a + b + ab

. After 99 moves, one number remains. Find it.

Example 42

challenge

7\times 7

board has

49

lights, all off. A move toggles every light in one row OR every light in one column. Can you reach a configuration with exactly one light on?

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Related Concepts

Equal Algebraic Invariance

Background Knowledge

These ideas may be useful before you work through the harder examples.

equal