In the 12 days of X-Math, our four little monsters take on their own holiday preparations at the North Pole. Each day leading up to Dec. 25, we'll reveal a new puzzle to solve — and slice of this seasonal story.

Follow along below to uncover all the festive fun (and havoc) the holidays have in store! Can you solve all 12 days of our holiday tale?

Answer key is posted at the end of the article!

## 12 Days of X-Math

### Day 12

It’s a massive snowstorm! Five inches of snow is falling every hour. How many days will it take for the snow to reach this regulation-height basketball hoop?

### Day 11

Nine degrees is very cold, but 90 degrees is just right. In this diagram of a stocking, there are lots of right angles. How long is the missing length, in inches?

Here's the schematic in full:

### Day 10

Grogg is playing a guessing game with 10 stockings that all look the same. 5 stockings hide candy, and 5 hide coal. (Fortunately, Grogg likes to eat both.)

He must guess whether the leftmost stocking hides candy or coal. Once it’s revealed, he can guess at the next stocking. This continues for all 10 stockings.

Using their best strategy to guess correctly, the probability that Grogg guesses all 10 stockings correctly is 1/D. What is the sum of the digits of D?

### Day 9

12 friends have hats placed on their heads. Each hat is red or green. They are told that, among them, there is at least one red hat and one green hat. They can see everyone else’s hat, but never their own.

Beaker, who is wearing a green hat, says: “I have no idea what color my hat is.”

Grogg immediately says: “With that new information, I know what color my hat is.”

How many red hats are being worn?

### Day 8

Nellie has 2 different options for lights, solid and blinking. She will set up 5 lights in a row, using at least one of each type. Nellie also decides it’s not cool if there are two solid lights next to each other. So, she sets up every possible different cool light display at her friends’ homes. How many friends now have a cool light display?

### ‍Day 7

Lizzie is building a right triangle where the longest leg has length 7. If Lizzie wants the length of the hypotenuse to also be an integer, how many different options are there for the length of the hypotenuse?

### Day 6

Grogg gets 11 pine treats. He starts thinking of ways to write numbers with just 1s, and writes down this expression. What is it equal to?

### Day 5

The Beast Store only offers packs of 3 pine treats and 7 pine treats. What is the largest number of pine treats Winnie can’t buy using only these pack sizes?

### Day 4

If 8 reindeer can eat 12 pine treats in 8 minutes, how many pine treats can 4 reindeer eat in 4 minutes?

### Day 3

Lizzie sees 6 sweaters in this mysterious system of equations. What number is represented by the reindeer?

(sweater) + (sweater) + (sweater) = 18

(sweater) + (reindeer) + (pine tree) = 17

(sweater) + (reindeer) + (stocking) = 19

(sweater) + (pine tree) + (stocking) = 14

### Day 2

Alex has 4 miles worth of yarn to make sweaters. Looking online, Alex is shocked to find out that it takes 1100 yards of yarn to make one sweater for one child. How many complete children’s sweaters can Alex make with his yarn?

### Day 1

One day, Grogg traveled to the North Pole. Starting from the North Pole, Grogg went 4 miles south, then 3 miles east. How far away is Grogg from the North Pole?

We actually shared all the answers already. Hint: Just look at the bold print! The answer to each problem is in the bold text of the next day's problem. Below are all of the days' answers in one place, along with some explanations.

Day 1

4 miles, not 5, because it’s the North Pole.

Day 2

6 sweaters. The total is 6.4 — but .4 of a sweater won't keep you warm, especially at the North Pole!

Day 3

The reindeer in the equation represents 8.

Day 4

They can eat 3 pine treats in 4 minutes.

Day 5

Winnie can't buy 11 pine treats using those pack sizes.

Day 6

The answer is 7, no units (TeX source: https://artofproblemsolving.com/texer/qvoeuxbr)

Day 7

There are 2 different options: the hypotenuse must be shorter than 7√ 2, which is less than 10, and it must also be longer than 7. The only options are 8 and 9, both possible (√(15 - 7 - 8) and 4√( 2 - 7 - 9)).

Day 8

12 friends, which include SBSBS, SBSBB, SBBSB, SBBBS, SBBBB, BSBSB, BSBBS, BSBBB, BBSBS, BBSBB, BBBSB, BBBBS.

For N options this is typically F_{n+2}, a Fibonacci number, but the requirement that there is at least one of each color removes BBBBB as the 13th option.

Day 9

There are 10 red hats; Grogg knows their hat is green because they see only one green hat on Beaker. And if Beaker didn’t see any green hats, they would know their own hat is green.

Day 10

The sum of the digits of D is 9. The probability is 1 in 252.

Day 11

The missing length is 5 inches.

Day 12

It will take one day. One day — 5 inches x 24 hours = 120 inches = 10 feet, the height of a regulation basketball hoop.

Want more challenges like these? Learn more about our live online classes at AoPS Online and Virtual Campus, or in-person classes at a campus near you.
For more adventures with our little monsters, check out our self-paced, comic-based elementary curriculum at Beast Academy.

#### Subscribe for news, tips and advice from AoPS

Thank you! Your submission has been received!
Oops! Something went wrong while submitting the form.
By clicking this button, I consent to receiving AoPS communications, and confirm that I am over 13, or under 13 and already a member of the Art of Problem Solving community.