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

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Show that the sum โˆ‘i=1nai\sum_{i=1}^n a_i is invariant under any permutation ฯƒ\sigma of the indices {1,2,โ€ฆ,n}\{1,2,\dots,n\}.

Example 2

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A lamp toggles ON/OFF each press. After 20252025 presses starting from OFF, what state is it in, and which invariant tells you?

Example 3

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

Example 4

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A frog on the integer line at 00 can jump ยฑ2\pm 2 or ยฑ5\pm 5. Which positions can it reach?

Example 5

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In the equation 4(xโˆ’1)=124(x-1) = 12, divide both sides by 44. What is invariant and what is the result?

Example 6

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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

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Numbers 1,2,3,4,51,2,3,4,5 are on a board. You repeatedly replace two numbers by their sum. Is the sum of all numbers on the board invariant?

Example 8

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

Example 9

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On a board are the numbers 1,2,3,โ€ฆ,1001, 2, 3, \dots, 100. You may replace any two numbers a,ba, b with โˆฃaโˆ’bโˆฃ|a-b|. After 99 such operations, one number remains. Is its parity determined, and if so, what is it?

Example 10

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

Example 11

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You replace x=7x = 7 with x+4=11x + 4 = 11. Is the solution set invariant?

Example 12

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A token sits at 00 on a number line; each move adds 33 or subtracts 33. What is invariant about its position mod 33?

Example 13

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Three jars contain a,b,ca, b, c liters of water with a+b+c=9a+b+c = 9. A move: pick two jars and equalize them (each gets the average). Find an invariant.

Example 14

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Numbers 1,1,2,3,5,81, 1, 2, 3, 5, 8 are on a board. You may replace any two numbers a,ba,b with aโ‹…ba \cdot b. Is the product of all numbers on the board invariant?

Example 15

challenge
Numbers 1,2,โ€ฆ,10241,2,\dots,1024 are on a board. Repeatedly erase two numbers a,ba,b and write โˆฃaโˆ’bโˆฃ|a-b|. After 10231023 steps one number remains. Show its parity is determined, and find that parity.

Example 16

easy
Fill in the blank: under the map xโ†ฆx+360โˆ˜x \mapsto x + 360^\circ, the value of cosโกx\cos x is ____.

Example 17

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You add 55 to both sides of x=3x = 3. What is invariant?

Example 18

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Is the perimeter of a rectangle invariant when you scale it by a factor of 22?

Example 19

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Under the map nโ†ฆn+4n \mapsto n + 4 on integers, what is invariant about n(mod4)n \pmod{4}?

Example 20

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What is invariant when you multiply both sides of 2x=62x = 6 by 12\frac{1}{2}?
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