### Brain Teasers

Each week, we ask a brain teaser in The Commons Digest newsletter. Here, you'll find all of the answers!

Sign up for the newsletter HERE.

#### Week 29

If you have seven white socks and nine black socks in a drawer, how many socks do you have to pull out blindly in order to ensure you have a matching pair?

Answer: Three socks. If the first one is one color, and the second one is another color, then the third sock will make a matching pair, regardless of what color it is. Here are the possibilities after pulling out three socks:

1. All 3 black

2. All 3 white

3. 2 black 1 white

4. 2 white 1 black

Tip: This is a good reminder to not overcomplicate your answer!

#### Week 28

An apple costs 40 cents, an apricot costs 60 cents, and a grapefruit costs 80 cents. How much does a pear cost?

Answer: You're trying to find a pattern. One answer might be rooted in discovering that each vowel = 20 cents. An apple has two vowels = 40 cents. Apricot has 3 vowels = 60 cents. Grapefruit has 4 vowels = 80 cents. Therefore, a pear has 2 vowels = 40 cents.

#### Week 27

A man pushes his car to a hotel and tells the owner he’s bankrupt. Why?

Answer: He's playing Monopoly!

#### Week 26

*Skipped for an end of year community celebration!*

#### Week 25

I bought a cow for $800. I sold it for $1000. I bought it again for $1100. I sold it again for $1300. How much did I earn?

Answer: $400. Remember to think of this like a stock purchase.

- You bought it for $800 and sold it for $1000 = $200 earnings
- Then you had a separate purchase at $1100 and sale at $1300 = $200 earnings
- Total earnings = $400

#### Week 24

A boy and a girl are sitting on a bench. “I’m a girl,” says the child with brown hair. “I’m a boy,” says the child with blond hair. If at least one of them is lying, which one is lying?

Answer: Both are lying. If any of them told the truth, they would deliver one same answer.

#### Week 23

If a zookeeper had 100 pairs of animals in her zoo, and two pairs of babies are born for each one of the original animals, then (sadly) 23 animals don’t survive, how many animals do you have left in total?

Answer: 977 animals. You need to carefully read the wording here to follow along with all of the "pairs". To start, she has 100 pairs of animals = 200 animals at the zoo. Two pairs of animals are born for each of the original animals. Two pairs = four babies, per original animal. There were 200 original animals, so 200 X 4 = 800. Adding the 800 to the original 200 = 1000 total animals. 23 animals do not survive, so that leaves us with 1000 - 23 = 977. This one is easy as long as you take your time!

#### Week 22

I left my campsite and hiked south for 3 miles. Then I turned east and hiked for 3 miles. I then turned north and hiked for 3 miles, at which time I came upon a bear inside my tent eating my food! What color was the bear?

Answer: White. You may be thinking, how the heck should I know? Or you might take a random guess like black or brown. But, the answer is always hidden in the clues...the only place where you can walk in a triangle (vs a square) and end up back in the same spot is the north pole. What types of bears live at the north pole? Polar Bears. They're white.

#### Week 21

What number should replace the question mark?

*Answer: *Two. In each box, the bottom left number + the top left number = the top right number. The bottom right number plus the bottom left number = the top left number. So, 2 + 4 = 6. 2 + X = 4, therefore it must be 2.

#### Week 20

The alphabet is written here but some letters are missing.Arrange the missing letters to give a word.What is it?

ABDFGHIJKLMPRVWXYZ

Answer: Conquest

#### Week 19

What number and letter continues this sequence?

12 T 17 E 22 F 27 N ? ?

Answer:

32 and F. First, calculate the difference in the numbers to see if there's a pattern. They increase by give. That means 27 + 5 = 32.

For the letters, they represent the first letter of the number, when the two digits are added together. For example: 1 + 2 = three = T. 1 + 7 = eight = E. 3 + 2 = five = F.

#### Week 18

Question: Michelle’s mom has four children. Her first child is named April, her second child is named May and her third child is named June. What is the name of her fourth child?

Answer:

Michelle. If you got that wrong (many people say July!), re-read the prompt slowly. This is really a test of listening skills, logic and quick thinking.

#### Week 17

Question: You have an 8 X 8 X 8 cube made up of 1 X 1 X 1 cubes. If you fully dip the cube in paint, how many 1 X 1 X 1 cubes are coated on 3/2/10 sides?

Answer:

First, calculate how many squares are on the cube. A 8 x 8 x 8 = 64 X 8 = 512 squares. Since each square is made up of 1 X 1 X 1 cubes, you have 512 1 x 1 x 1 cubes.

You can either answer by elimination, like this:

**Coated on 3 sides:**the only cubes that can be coated on three sides will be the corner cubes. You'll have 8 corner cubes. That means you have 512 - 8 remaining cubes.******Coated on 2 sides:**Only the edge cubes will be coated on two sides. There are 6 cubes are along each edge and there are 12 edges. So 6 x 12 = 72 cubes that must be coated on 2 sides. That means we have 504 – 72 = 432 cubes remaining.******Coated on 1 side:**The cubes that will be coated on one side are the "face" cubes. 6 x 6 cubes are on each each face, once you exclude edges and corners. Since there are 6 faces, you'll take 36 X 6 = 216 cubes that are coated on 1 side. 432 – 216 = 216 cubes remaining.*Therefore 216 cubes are coated on 0 sides.*

Or you can get straight to it:****

**Coated on 0 sides:**6 x 6 x 6 cubes are not on the faces, edges or corners. 36 x 6 = 216 are coated on 0 sides, which matches up to the above answer from elimination

#### Week 16

Question: A farmer wants to cross a river with his fox, chicken, and corn bag. The boat he intends to use can only fit the farmer and one other thing. His fox and chicken are hungry, so if he leaves them together the fox may eat the chicken. Also, if he leaves the chicken with the bag of corn, the chicken might eat it. How can the farmer take everything across the river safely?

Answer: The farmer should take the chicken first, so the fox doesn't eat it. Then, he can go back and take the fox across. When he gets to the other bank, he should drop the fox and take the chicken across to get the corn (so the fox doesn't eat the chicken). Then, he should drop the chicken and take the bag of corn across. Finally, the farmer should go back to the starting location for the chicken.

#### Week 15

Question: You are given 12 balls. 11 of them are identical and 1 of them is a little bit heavier. If you can use the scale only 3 times, how can you determine which ball is heavier? Note that the scale compares two weights against each other.

Answer:

First, split the balls into three groups of four. Weigh one set (say 1,2,3,4) against the next set (5,6,7,8). One of two things will happen:

Scenario 1: Both sets weigh the same. That means that you know the heavier ball has to be in the final, unweighed group: 9,10,11,12.

Scenario 2: One of the first groups is heavier than the other. The heavier group contains the heavier ball.

Next, take whatever group it is that you know is heaviest. Say it is 9,10,11,12. Split that group into two, and measure 2 balls against two balls (say 9 and 10 vs. 11 and 12. One set will be heavier than the other.

As your third measure, weigh the two balls from the heavier side against each other. The heavier one is your heaviest ball.

#### Week 14

Question: Given two hourglass of 4 minutes and 7 minutes, the task is to measure 9 minutes.

*Answer:*

- At 0 minutes: Start both hourglasses at the same time.
- At 4 minutes: The 4 min hourglass will have run out. The 7 min hourglass will have run for 4 minutes, leaving 3 minutes of 'sand' left to run.
- Flip the 4 min hourglass again.
- At 7 minutes: The 7 min hourglass is going to run out next, as it only had 3 minutes left in it. At this point, the 4 min hourglass has also been running for 3 mins, so there is 1 min left in that hourglass.
- Flip the 7 min hourglass again.
- At 8 minutes: The 4 min hourglass is going to run out after a minute, bringing us to 8 minutes. The 7 min hourglass still has 6 minutes left in it (i.e. 1 minute remaining)
- At 9 minutes: the 7 min hourglass becomes empty!

#### Week 13

Question: A man is allocated a task. He doubles the task done everyday. If the man completely does the task in 18 days, how many days did it take for the man to complete 25% of the task?

*Answer: Day 16.*

**It's best to work backwards for these types of questions. On day 18 he's completed 100% of the tasks. Since he doubles his output each day, we know that he must have completed 50% of the tasks on day 17 (one day prior to day 18). The day before that - day 16 - would have to be half of day 17. Half of 50% (day 17) would be 25% of the total.

#### Week 12

Question: You have got someone working for you for five days and a gold bar to pay him. You must give them a piece of gold at the end of every day. What are the fewest number of cuts to the bar of gold that will allow you to pay him 1/5th each day?

*Answer: *Two cuts. Say you make two cuts to the gold bar and end up with three pieces:

- One that is 1/5th of the bar (we'll call that one payment unit)
- Two that are 2/5ths of the bar (two payment units)

Here's how you would pay out the worker:

- Day one, the worker starts with 0 bars and you pay him the first segment of payment unit
- On day two, he now already has one payment unit. You give him the segment with two payment units and ask him to give you back the one payment unit.
- On day three, he now has one block worth of two payment units. You give him the one payment unit back to pay him for day three. You don't ask for anything in return.
- On day four, you pay him your final segment of two payment units and ask for the one payment unit in return. He now has two segments, each worth two payment units (totalling four payment units).
- On day five, you give him the segment with one payment unit. He now has all of the segments and was paid 1/5th of the bar each day!

#### Week 11

Question: You have 9 chairs, 7 clients and me. What are the chances you're sitting next to me?

*Answer: *Given that the interviewer is sitting in one of the 9 chairs, the other 8 chairs are distributed randomly to the 7 clients and you. Out of the 8 chairs you could possibly sit in, there are 2 combinations where you are sitting next to the interviewers (i.e. the 2 chairs on either side of them).

Thus, the chance that you are sitting next to the interviewer is 2/8 or 25%. Which chair the interviewer sits in is irrelevant in this question, given that every person has an equal probability of sitting in every seat (1/9). How the rest of the clients sit is also irrelevant, as without further information it is logical to assume the clients are identical and have no preference on a specific seating arrangement.

#### Week 10

Question: You are blindfolded and are sitting in front of a table. There are 100 coins on the table, 50 of them face up and 50 of them face down. How do you split them into 2 groups where both groups have the same number of coins face up and face down? You cannot feel the coin faces.

*Answer: Split the coins into two groups of 50 coins each, by counting them out at random. Flip one of the groups over so that the coins that were facing down are now facing up, and vice versa for the coins that were facing up. Both groups should now have the same number of couns face up and face down, since there were an even number of each. Say for example, when you split the coins originally, the left pile had 40 up and 10 down. That means that the right pile had to have 40 down and 10 up (since there are even amounts of each). Flipping the right pile would mean you now have 40 up and 10 down, matching the left pile.*

#### Week 9

Question: There are three boxes of eggs. In each box there is either a set of big eggs, small eggs or big and small eggs mixed. The boxes are labelled (shock) LARGE, SMALL and MIXED but each box is labelled incorrectly. What is the least number of boxes you can open to know which eggs are in which box and why?

*Answer: *One!

You need to remember that *all* boxes are labelled incorrectly.

Here's an example.

- You open the box marked BIG and say it's filled of the small eggs. You now know that one boxes must contain the big and mixed eggs. Which box is which? Well, one of the boxes was labelled SMALL, but you opened that box with #1. That means that the third box must have been labelled MIXED, since it's the only one left. You also know that they're all labelled incorrectly, so the MIXED label cannot be the mixed eggs. That means those eggs must be big.

#### Week 8

Question: At exactly noon each day a scientist puts a bacteria in a petri dish. Every minute the bacteria divides in two. When it's at 1pm, the dish is full. What was the time when the dish was half full?

*Answer: *This brain teaser is a good reminder that you don't always have to overcomplicate things!

After all, the answer is quite simple. If every minute the bacteria doubles, and it’s full at 1pm. It would have been half full a minute earlier at 12:59pm.

#### Week 7

Question:

There are three boxes, one contains only apples, one contains only oranges, and one contains both apples and oranges. The boxes have been incorrectly labeled such that no label identifies the actual contents of its box. You can pick one box to open (knowing what it has on the label). Without looking in the box, you take out one piece of fruit. By looking at the fruit, how can you immediately label all of the boxes correctly?

*Answer: *

You know all 3 boxes are incorrectly labeled. You want to avoid pulling fruit out of the box with both apples and oranges, since you won't know if it's the both with just the one fruit or both fruits.

So, open the box labeled “Apples + Oranges.” Since the label is wrong, it must have one or the other not both.

Whichever fruit comes out is what that box contains. If you took out an apple, the box has only apples. If you took out an orange, the box has only oranges.

Let's say you pull out an apples. You can then move the Apples label to this box. The box that had the apples label has either oranges or both fruits. The third box will have whatever is left (again, either oranges or both fruits). The third box must have had the "oranges" label. And since all boxes are labeled incorrectly, we know that oranges can't be in box #3. So move the oranges label to box #2 and the Apples + Oranges label to box #3.

#### Week 6

Question:

Four investment bankers need to cross a bridge at night to get to a meeting. They have only one flashlight and 17 minutes to get there. The bridge must be crossed with the flashlight and can only support two bankers at a time.

The Analyst can cross in one minute, the Associate can cross in two minutes, the VP can cross in five minutes, and the MD takes 10 minutes to cross.

How can they all make it to the meeting in time?

*Answer*:

First crossing: the Analyst takes the flashlight and crosses the bridge with the Associate = 2 minutes

Return crossing: the Analyst returns with the flashlight = 1 minute

Second crossing: The Analyst gives the flashlight to the VP and the VP and MD cross together = 10 minutes

Return crossing: The VP gives the flashlight to the Associate (since the Associate only needs 2 minutes) = 2 minutes

Third crossing: The Analyst and Associate cross the bridge again together = 2 minutes.

Total Time = 17 minutes

#### Week 5

Question:

Let’s say that you have 25 horses, and you want to pick the fastest 3 horses out of those 25. In each race, only 5 horses can run at the same time because there are only 5 tracks. What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?For the sake of clarity, let’s assume that each horse’s speed doesn’t change regardless of race conditions, fatigue or good ol’ fashioned determination.

You can consider the horses as mechanical, and programmed to run the same speed each time they race. With that said, the horses will come in the same place every race. You can never know their times, only what place they finish in.

How many races will it take to find the top three horses?

*Answer: Check out **Matt's article** for great visualizations. *

Step 1: Race every horse at least once. Why? Any horse left out could be one of the top three. Since there are 25 horses, and 5 horses per race (25/5) that means **we need to run 5 races to start. **

Step 2: You now know which horse was the fastest in each race. But that doesn't mean that they're all in the top three. For example, there could be up to three horses from one race that were faster than all other horses - even the lead horse - in the other races. So, we need to race the winning horses to understand how each of the races stacks up against each other. **Therefore, we're now at race 6.**

Step 3: We can now order each of the original five races, base don the placement of each of the lead horses in race 6. We still have the predicament that there could be multiple horses in one of the original five races that makes the top 6. So how do we unpack that? Let's order the races as follows:

- Group 1 = horses from the first group of 5 races that had the fastest horse from race 6
- Group 2 = horses from the first group of 5 races that had the second fastest horse from race 6, etc.
- Group 3 = etc.
- Group 4 = etc.
- Group 5 = horses from the first group of 5 races that had the slowest horse from race 6

Since we're only looking for the top 3 horses, we can eliminate a number of horses based on these facts:

- We
*know*that all of group 4 and all of group 5 were slower than the leading horses from group 1, 2 and 3. Therefore all of group 4 and 5 can be eliminated (we don't need to race them again) - We
*know*that the rest of group 3 (outside the lead horse) will not be in the top 3 either - We
*know*that for groups 1 and 2, only the 3 fastest horses will be in the running (meaning we can eliminate the horses that run in positions 4 and 5 from those groups)

That leaves us with 6 horses:

- The lead horse from group 1
- The 2nd and 3rd place horses from group 1
- The lead horse from group 2
- The 2nd and 3rd place horses from group 2
- The lead horse from group 3

However, we can only run a race with 5 horses. So what do we do? We can eliminate the lead horse from group 1 since we already know it's the fastest horse. At this stage, we're only looking for the second and third placed horses.

**We now run our seventh race with the remaining 5 horses. **

Final Answer: 7 races!

#### Week 4

Question: How many regular golf balls can you fit into a plane?

*Answer: Well, I am sure you guessed, there's no specific answer here. Instead, the interviewer is looking to understand how you think and how you tackle an ambiguous problem. For example:*

*How do you estimate how large the plane is (do you estimate based on what you believe to be the most common plane size? If so, how do you determine that?)**Do you remember to flag, and remove space, for fixtures like seats?*- Do you consider the passenger area, cockpit, overhead and stowaway compartments?
*What sort of educated guesses are you making along the way?**Are you able to tackle this effectively without getting too granular (80/20?**Tip: think out loud the entire time!!*

#### Week 3

Question: What is the next number in the following sequence: 0 0 1 2 2 4 3 6 4 8 5?

*Answer: 10. There are two alternating sequences: 0, 1, 2, 3, 4, 5 and 0, 2, 4, 6, 8.*

#### Week 2

Question: You need to measure out four gallons of water, but you only have a three-gallon jug and a five-gallon jug. How do you measure out four gallons exactly?

*Answer: Fill the three-gallon jug and pour all of the water into the five-gallon jug. Then, fill the three-gallon jug again and use it to continue filling the five-gallon jug. Since the five-gallon jug already had three-gallons of water, there are only two-gallons of space remaining in the five-gallon jug. Therefore, two gallons will be poured out of the three-gallon jug, leaving one gallon in the three-gallon jug. Now, dump all of the contents out of the five-gallon jug. Pour the one gallon that is left in the three-gallon jug into the five-gallon jug, and then fill the three-gallon jug again and pour it into the five-gallon jug. This will give you four gallons.*

Looking for more interview prep? Join The Commons' Strategy + Operations Sprint to gain reps working on a business problem, doubling as interview prep. Plus, you'll get 1:1 access to mentors who live and breathe these jobs daily.

#### Week 1

Question: You are in a room that has three switches and a closed door. The switches control three light bulbs on the other side of the door. Once you open the door, you may never touch the switches again. How can you definitively tell which switch is connected to each of the light bulbs?

*Answer: Switch on switches 1 & 2, wait a few minutes. Then switch off number 2. Enter the room. Whichever bulb is on is wired to switch 1, whichever bulb is off and hot is wired to switch number 2, and the third is wired to switch 3 (this bulb should be off and cold).*