[Melissa Zhang]> Puzzles for K.

A genus one surface with one boundary component and a Jucys-Murphy braid. The following are some puzzles I worked on with an awesome middle schooler during the first year of the pandemic.

Algebra Puzzles

Puzzle 23
There are twice as many chickens as cows, and five more goats than chickens. If there are 15 goats, how many cows are there?
Puzzle 22
There are 36 chicken nuggets to be shared among 10 people. Each child eats 4 nuggets and each adult eats 3. How many children are there?
Puzzle 21
Tamara has 35 coins in nickels and quarters. In all, she has $4.15. How many of each kind of coin does she have? [Source]
Puzzle 20
There are 17 people at a picnic and each is eating either a hotdog or hamburger. There are three more hotdogs than hamburgers. How many hamburgers are there?
Puzzle 19
Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore; how much did she spend there? [Source]
Puzzle 18
There are 15 ducks at the pond. The number of ducks is three more than four times the number of swans. How many swans are there? [Source]

Summer Puzzles

Puzzle 17
How many times each day (24 hours) do the hour hand and minute hand of a clock cross?
Puzzle 16
You have a half gallon bottle of orange juice (8 cups). You also have two empty bottles: one holds 3 cups, and the other holds 5 cups. Your friend asks you for exactly 4 cups of orange juice in a bottle. How do you do this without wasting any juice?
Puzzle 15
Your eyes are closed (no peeking) when I pour 100 pennies on the table. I tell you that exactly 20 of them are heads up. Now, with your eyes still closed, you have to make two piles, one your left, and one on your right, such that the two piles have the same number of heads. You can't tell whether a coin is heads up or tails up by feeling or anything. You are allowed to move coins around and flip them as much as you want. How do you do it?
Puzzle 14
You meet twin brothers B1 and B2, guarding doors D1 and D2. You know that one of them always tells the truth, and one of them always lies. You also know that one of the doors leads to a trash can, while the other leads to a large pile of money. You are allowed to ask just one of them exactly one question, and then you have to pick a door and walk through it. What do you ask?
Puzzle 13
At a summer camp, there are three bullies, A, B, and C, who each have one little sibling, a, b, and c, respectively. The six of them come to a river and find a two-person boat. If a little sibling is on the boat with a bully who is not their older sibling, then they will get picked on. (There are camp counselors on the banks though, so no one gets picked on except possibly on the boat.) How do you make sure everyone gets across the river, without anyone getting picked on?
Puzzle 12
This is also a classic puzzle.
You have 12 visually identical coins. You know that 11 of them are real and 1 of them is fake. The fake coin might be heavier or lighter than the real ones. Using a balancing scale no more than 3 times, can you figure out which one is the fake coin?
Puzzle 11
This is a folklore puzzle that's been around for centuries.
A man has pet wolf, a pet goat, and a pet cabbage. The wolf really wants to eat the goat, and the goat really wants to eat the cabbage, but they won't do anything bad in front of the man. (The wolf also doesn't like the taste of cabbage.) The four of them come to a river, and the man rents a boat. The boat can only carry two of them at a time (the cabbage is really, really big). He's the only one than can row the boat. How should he shuttle everyone across the river safely?
Puzzle 10
I learned this puzzle from one of Professor Mo Hendon's emails.
There are two people, conveniently named P and S. Someone chooses two single digit integers 0 < a,b < 10 and tells P their product p = ab, and tells S their sum s = a+b. P and S are infinitely logical, and they know everything stated in the puzzle so far. The following conversation ensues:

P: I don't know the two numbers.
S: I don't know the two numbers.
P: I don't know the two numbers.
S: I don't know the two numbers.
P: I don't know the two numbers.
S: I don't know the two numbers.
P: I don't know the two numbers.
S: I don't know the two numbers.
P: Now I know the two numbers.

What are the two numbers?
Puzzle 9
Source: The Math Forum (mathforum.org)
Dee finds that she has an extraordinary social security number. Its nine digits contain all the digits from 1 to 9. They also form a number with the following characteristics:

a. When read from left to right the first two digits form a number divisible by two.
b. The first three digits form a number divisible by three.
c. The first four digits a number divisible by four, and so on, until the complete number is divisible by 9.

What is Dee's social security number?
Puzzle 8
Can you arrange the numerals 1 to 9 (1, 2, 3, 4, 5, 6, 7, 8 and 9) in a single fraction that equals exactly 1/3 (one third)? An example that doesn't work: 7192/38456 = 0.187
You can use a calculator, but you'll need to think about what kinds of fractions equal 1/3 first.
Puzzle 7
I heard this puzzle from Kevin Sackel.
You have a 3"x3"x3" block of cheese that you need to chop into 1"x1"x1" (cubical) serving portions. You only have a straight kitchen knife and a cutting board. What is the minimum number cuts you'll need to make? How do you know?
Puzzle 6
You and I are sitting around a circular table, and are about to play the following game. We take turns putting quarters on the table. The quarters can't touch, and they can't hang off the edge of the table. We play until we can't put any more quarters on the table. The person who puts the last quarter on the table gets to keep all the money. I let you choose whether you want to go first or second. Which should you choose, and what is your strategy? Can you make sure you win?
Puzzle 5
You and three of your friends are hiking at night and come across a rickety bridge with a lot of missing planks. You have 17 minutes to get everyone across to the other side. There is only one flashlight, and the bridge can only hold two people at a time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth; it cannot be thrown, etc. Each of you walks at a different speed. A pair must walk together at the rate of the slower person.

Usain: 1 minute to cross
K: 2 minutes to cross
Alice: 5 minutes to cross
Bob: 10 minutes to cross

For example, if Usain and Bob walk across first, 10 minutes have elapsed when they get to the other side of the bridge. If Bob then returns with the flashlight, a total of 20 minutes have passed, and you have failed the mission.
Puzzle 4
[Source] A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one.

The 1st student said, "The number is divisible by 2."
The 2nd student said, "The number is divisible by 3."
The 3rd student said, "The number is divisible by 4."
(and so on)
The 30th student said, "The number is divisible by 31.

The teacher then commented that exactly two students, who spoke consecutively, spoke wrongly.

Which two students spoke wrongly?
Puzzle 3
[Source] Three people rent a room at $30. They pay $10 each and go up to the room. The owner realized he charged too much and it was only supposed to be $25. He sends the bell hop up with the $5. Each of the people keeps $1 and they give the bellhop $2 as they can't share it. So now each person has paid $9 for the room (total $27) and the bell hop has $2... where is the other $1? (What's going on?!?)
Puzzle 2
Carly and Caleb are driving bumper cars toward each other on a straight track. Carly is driving at 1 ft/s and Caleb is driving at 2 ft/s. When they're 60 ft away from each other, a fly that was on Carly's front bumper starts flying toward Caleb's front bumper. When it lands on Caleb's front bumper, it turns around the flies back toward Carly's bumper, and continues back and forth until Carly and Caleb bump into each other. The fly's speed is 5 ft/s. How much distance does the fly fly before it gets squished between Carly and Caleb's bumpers? (Sorry about what happened to the fly in this riddle.)
Puzzle 1
For all the numbers from 1--100, write down an expression using only four '4's (and no other digits) that is equal to that number. For example, 1 = (4*4)/(4*4). I'll also do this and we can compare which numbers we thought were the hardest, and see if we come up with different solutions. (You can start by just trying the first 20 to see if it's fun.) Technically, the rules are that you can only use four 4s and addition, subtraction, multiplication, division, decimal point (e.g. 10 = 4/.4 + (4-4)), and factorial. But if you come up with something interesting using another operation, I'd be interested to see!