# Toilet Paper Math Puzzles

## Toilet Paper Math Word Problems

Toilet paper can be used for so much more than cleaning your bottom after a sweet release. You can decorate your neighbor's lawn with TP, make mummy costumes, curl your hair, and irritate your roommates to no end by installing the roll backwards. Toilet paper can also inspire a lot of math word problems. Next time you're on the white throne, keep busy with these easy and challenging toilet paper math puzzles. You can even use your TP as scratch paper.

## #1 Toilet Paper Usage (Grades 4 to 6)

Kelly uses 11 squares of toilet paper for "number one", and 26 squares for "number two." She goes number one 7 times a day, and number two every other day. If her brand of toilet paper has 240 squares per roll, how many rolls does she use in four weeks?

## #2 Toilet Paper Economics (Grades 4 to 7)

Sinclair's favorite brand of toilet paper comes in different sizes. The jumbo roll has 25% more paper than the regular roll, and the ultra roll has 20% more paper than the jumbo roll. The store sells a 40-pack of ultra rolls for $57, a 24-pack of jumbo rolls for $32, and a 30-pack of regular rolls for $30. Which package is the best value in terms of amount of paper per dollar? Which package is the worst value?

## #3 Measuring with Toilet Paper (Grades 4 to 8)

Katie measured the height of her younger brother and older brother with toilet paper and found that her older brother was twice as many squares tall as her younger brother. When she measured herself, she found that she was 3 TP squares taller than her younger brother. If the total of all three siblings' heights is 39 TP squares, how tall are the three siblings (in TP squares)?

## #4 Toilet Paper Scrolls (Grades 5 to 7)

Dan is transcribing the Bible onto rolls of toilet paper. On every square he writes exactly 135 words. If his copy of the Bible has 783,478 words, and his toilet paper has 162 squares per roll, how many rolls of toilet paper does he need to complete his project?

## #5 Toilet Paper Perimeter and Area (Grades 5 to 10)

If a roll of toilet paper is 72 feet long, what is the area of the largest triangle you can enclose with the whole length of toilet paper, i.e., what is the largest area of a triangle with a perimeter of 72 feet? What is the area of the largest rectangle you can enclose with the whole length of toilet paper? What is the largest area you can enclose with the toilet paper, using any shape?

## #6 Thickness of Toilet Paper (Grades 8 to 12)

A roll of toilet paper is 233 meters long. When it is wrapped around a tube with a diameter of 4 cm, the entire roll has a diameter of 30 cm. Assuming the paper is not a fluffy quilted type of toilet paper, and it is wrapped as tightly as possible, how thick is the toilet paper (in cm)?

## #7 Toilet Paper Pyramid (Grades 11 and Up)

Ken builds a triangular pyramid with rolls of toilet paper. Counting from the top, the first level has 1 roll, the second level has 3 rolls, the third level has 6 rolls, and so on, so that the Mth level has M(M+1)/2 rolls (the Mth triangular number).

Ken then dismantles the triangular pyramid to build a square pyramid with the rolls instead. Counting from the top, the first level has 1 roll, the second level has 4 rolls, the third level has 9 rolls, and so on, so that the Nth level has N^2 rolls (the Nth square number).

When building his square pyramid, he discovers that he has one roll left over, i.e., the square pyramid requires one fewer roll than the triangular pyramid. If Ken's pyramids contain more than 100 rolls, what are the values of M and N?

## #8 Toilet Paper Logic Problem (Grades 4 and Up)

Four friends (Alyx, Blake, Cagney, and Dev) went to the store to buy toilet paper for their respecetive homes in three different colors (white, pink, and black) and three different quantities (12-pack, 16-pack, and 20-pack).

- Two people bought white toilet paper and two people bought 12-packs.
- Dev bought darker paper than Alyx, who bought more than Blake, who bought lighter paper than Cagney, who bought more than Dev.
- The amount of paper Cagney bought divides evenly into the total amount that Alyx and Blake bought.
- For each friend, the total number of letters in the color and amount of TP is 11.

Can you figure out how much TP each person bought and what color?

## #9 Toilet Paper Roll Storage

Dustin never likes to run out of toilet paper, so he built a large rack to hold 400 cylindrical toilet paper rolls arranged in a 20-by-20 square, as shown in the image below. One day his girlfriend came over bearing the gift of toilet paper, but Dustin said his rack was full and couldn't fit another roll.

When Dustin wasn't looking, she rearranged the contents of the rack and discovered she increase the number of rolls by more than 10% and still have them all fit inside. She didn't stack any rolls on top of the rack, nor did she squish any rolls, remove their tubes or otherwise change their shape. How did she fit more rolls in the rack?

Here are the answers to the seven toilet paper math challenge problems with hints for solving.

**#1 -** Kelly uses 10.5 rolls in four weeks. To solve, find the number of times she goes number one and number two in a 28-day period. Then multiply each of these quantities by the number of TP squares she uses for each. Next, add the two amounts to get the total number of squares she uses in 28 days. Finally, divide this number by 240 to find how many rolls she needs.

**#2 -** The 40-pack of ultra rolls is the best value and the 24-pack of jumbo rolls is the worst value. To solve, notice that 1 jumbo = 1.25 regular, and 1 ultra = 1.5 regular. The 40-pack of ultra rolls is equivalent to 60 regular rolls, at a price of $57 this means each roll costs less than $1. The 24-pack of jumbo rolls is equivalent to 30 regular rolls, and at a price of $32 this means each roll costs more than $1. The 30-pack of regular rolls for $30 means each roll costs exactly $1.

**#3 -** Katie is 12 squares tall, her younger brother is 9 squares tall, and her older brother is 18 squares tall. To solve, you can either use guess-and-check, or set up an algebra equation. If the younger brother's height is X squares, then Katie's height is X+3 squares and her older brother's height is 2X squares. X + X + 3 + 2X = 39 gives the solution X = 9.

**#4 -** Dan needs 32 rolls to transcribe the Bible. To solve, first compute 162*135 = 24786 words per roll. Then divide 783478 by 24786 to obtain 31.61. Rounding up to the nearest whole roll gives 32. This is a calculator exercise.

**#5 -** An equilateral triangle with sides length of 24 produces the largest triangular area. A square with sides 18 produces the largest rectangular area. A circle with a circumference of 72 produces the largest area of any shape.

**#6 -** The toilet paper is 0.0298 cm thick, or equivalently 0.298 mm thick. To solve this problem, you need to know that the thickness of a stack or roll of paper is equal to the cross-sectional area divided by the total length. The cross-sectional area of a roll of toilet paper can be found by using the area formula for an annulus. See also How to Calculate the Thickness of Toilet Paper and How to Calculate the Thickness of Paper.

**#7 -** The base of the square pyramid is 9 rolls long (N = 9), while the base of the triangular pyramid is 11 rolls long (M = 11). The square pyramid uses 285 rolls, while the triangular pyramid uses 286 rolls. To solve this, you need to know that the sum of the first M triangular numbers is M(M+1)(M+2)/6 and that the sum of the first N square numbers is N(N+1)(2N+1)/6.

**#8 -** Alyx bought 20 rolls of white paper. Blake bought 12 rolls of white paper. Cagney bought 16 rolls of pink paper. Dev bought 12 rolls of black paper.

**#9 -** If Dustin's girlfriend increased the rack's capacity by more than 10%, then she fit at least 41 more rolls into the rack. There are several ways she could have done this. One solution, which invokes no extra assumptions (such as the diameter to height ratio of the rolls), is shown below. This arrangement contains 449 rolls, an increase of 12.25%.

## Comments

No comments yet.