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

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 2

easy

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

Example 3

easy

You change every

\$1

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

Example 4

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 5

easy

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

Example 6

easy

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

Example 7

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 8

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).

Example 9

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 10

easy

Is the perimeter of a polygon invariant under rotation?

Example 11

easy

A bag has

7

red and

3

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

Example 12

medium

A frog jumps on a number line by

\pm 3

each move, starting at

0

. Is its position mod

3

invariant?

Example 13

easy

You rearrange

3+5

into

5+3

. What is the invariant?

Example 14

easy

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

Example 15

easy

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

Example 16

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 17

easy

Two different objects both weigh

5

kg. Does sharing the invariant 'weight

=5

kg' make them identical?

Example 18

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 19

medium

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

Example 20

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?

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Related Concepts

Equal Algebraic Invariance