Practice Invariance in Math

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

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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.

Showing a random 20 of 50 problems.

Example 1

medium

Show that the sum

\sum_{i=1}^n a_i

is invariant under any permutation

\sigma

of the indices

\{1,2,\dots,n\}

Example 2

medium

A lamp toggles ON/OFF each press. After

2025

presses starting from OFF, what state is it in, and which invariant tells you?

Example 3

easy

Under reflection across a line, is the area of a triangle invariant?

Example 4

challenge

A frog on the integer line at

0

can jump

\pm 2

\pm 5

. Which positions can it reach?

Example 5

medium

In the equation

4(x-1) = 12

, divide both sides by

4

. What is invariant and what is the result?

Example 6

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 7

easy

Numbers

1,2,3,4,5

are on a board. You repeatedly replace two numbers by their sum. Is the sum of all numbers on the board invariant?

Example 8

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 9

medium

On a board are the numbers

1, 2, 3, \dots, 100

. You may replace any two numbers

a, b

with

|a-b|

. After 99 such operations, one number remains. Is its parity determined, and if so, what is it?

Example 10

easy

Under any rotation, is the distance between two points invariant?

Example 11

easy

You replace

x = 7

with

x + 4 = 11

. Is the solution set invariant?

Example 12

medium

A token sits at

0

on a number line; each move adds

3

or subtracts

3

. What is invariant about its position mod

3

Example 13

challenge

Three jars contain

a, b, c

liters of water with

a+b+c = 9

. A move: pick two jars and equalize them (each gets the average). Find an invariant.

Example 14

medium

Numbers

1, 1, 2, 3, 5, 8

are on a board. You may replace any two numbers

a,b

with

a \cdot b

. Is the product of all numbers on the board invariant?

Example 15

challenge

Numbers

1,2,\dots,1024

are on a board. Repeatedly erase two numbers

a,b

and write

|a-b|

. After

1023

steps one number remains. Show its parity is determined, and find that parity.

Example 16

easy

Fill in the blank: under the map

x \mapsto x + 360^\circ

, the value of

\cos x

is ____.

Example 17

easy

You add

5

to both sides of

x = 3

. What is invariant?

Example 18

easy

Is the perimeter of a rectangle invariant when you scale it by a factor of

2

Example 19

easy

Under the map

n \mapsto n + 4

on integers, what is invariant about

n \pmod{4}

Example 20

easy

What is invariant when you multiply both sides of

2x = 6

\frac{1}{2}

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Related Concepts

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