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We are given two glass jugs. Each contain the same volume V. One jug contains pure alcohol and second jug contains pure water. A modest quantity Q, of water is poured from the water jug into the alcohol jug. This content is thoroughly mixed and this diluted alcohol is added poured back into the water jug to equalize the volumes at their initial levels. What is the relationship between the final concentrations of alcohol in the alcohol jug and water in the water jug?
- When Q amount of water is added into the alcohol jug, the volume becomes V+Q. And the concentration of the alcohol change to
- [V/ (V+Q)]
- Now this diluted alcohol is added back to the water jug to equalize the volume.
- Concentration of the water becomes:
- [V / (V+Q)]
- Final concentration of Water and alcohol is the same.
- Another example:
- 1000 White marbels and 1000 Black mables
- Take 100 White marbles and add them to the bucket of 1000 black marbles.
- There are 1100 black marbles
- Mix them up
- Move 100 marbles back to the white bucket
- From this 100, there are 91 black marbles and 9 white marbles
- Final Concentration analysis:
- In the white bucket:
- 909 White mables, 91 Black Marbles
- In the Black bucket:
- 909 Black marbles, 91 White marbles
- In the white bucket:
- The final concentration is the same
Two bells. One ring five times a minute. 2nd one rings 4 times a minute. If they start at the same time, how long will be until they next ring together?
- 5 times a minute = every 12 seconds
- 4 times a minute = every 15 seconds
- Answer is 60 seconds. This is when the both bells will ring together.
What is the sum of integers from 1 to 100?
- n(n+1) / 2 = (100*101)/ 2 = 5050
An old style analogue clock falls off the wall and the face breaks into three pieces. The numbers on each piece add to the same total. Describe the three pieces
- Sum all clock’s digits: n(n+1)/2 = (12*13)/2 = 78
- Each piece has the sum of 26
- 11, 12, 1, 2 = 26 (easy)
- 5, 6, 7, 8 = 26
- 10,3, 9,4 = 26 (across)
12 marbles. A set of scale. All marbles weigh the same same, except one. This marble can be either light or heavier. You are given three attempts - determine which one is heavy/light.
- Create three gorups of 4
- A: 1, 2, 3, 4
- B: 5, 6, 7 8
- C: 9, 10, 11, 12
- Compare A and B
- Equal case is easy. Bad marble in C
- If A and B are not equal. Note which one is heavy
- Trick: Rotate
- A1, B5, B6, B7 vs. B8, C10, C11, C12
- Follow the same step as above
You are a bug sitting in one corner of a cubic room. You wish to walk to the extreme opposite corner. Describe the shortest path that you can walk?
- The shortest path between two points is called ‘geodesic’
- It is a cubic room. Open up the cube (like a cardboard box)
- You are trying to go from the bottom of cube to the top of the cube on the other side.
- Draw a straight line
- Use the pythogoream theorm to solve the problem:
- Sqrt(5)
Two cities: A and B. They are 1000 miles apart. There are 3000 apples in city A. Your goal is to deliver as many apples as possible. Caveat: Truck driver eats 1 apple per mile? How many apples can you deliver?
- It’s not ideal to take 1000 apples and drive all the way to the city B. You will have no apples left to deliver.
- Let’s agree that it is never a good idea to leave the town A without anything less than 1000 apples. Because since we are paying a cost of 1 apple per mile, we like to take as many apples we like.
- Optimal Strategy:
- Drop off location (when driver comes back - he is not eating our apples)
- We need to select a simple dump location. We will move forward in steps
- Problem is that you do not want to waste give extra apples free ride
- k - mile marker is the dump location
- N x (1000-k) = (N-1) x 1000
- solve for k
- k = 1000/N
- For the first time: N = 3 (b/c we are taking 1000 apples every time)
- Drop Apples at 333 Mark
- You will have 2001 apples after the third trip
- Optimize again:
- k = 1000/2 = 500 mark
- We are at 883 mark
- 1000 apples dumped at this location
- Optimize again:
- 1000 - 167 (miles left to travel)
- = 883 Apples delivered to City B
- 1000 - 167 (miles left to travel)
Picture 10x10x10 “macro-cube” flaoting in the air. The macro-cube is composed of 1x1x1 “micro-cubes”. Outer-layer falls off. How many micro-cubes are on the ground?
- 10^3 = 1000 Cubes to begin with
- Cube has six sides, and three of them opposite sides
- Since the cube from both side falls off, we should know that 8 cubes will remain (in between two sides)
- 8^3 = 2^3^3 = 2^9 = 512
- Answer: 1000 - 512 = 488
- Or Imagine:
- 1st side: 100 + 100 = 200 (falls off)
- 2nd side: 80 + 80 = 160 (falls off)
- 3rd side: 64 + 64 = 128 (falls off)
- Add them up: 488
How many degrees are there in the angle between the hour and minute hands of a clock when the time is qarter past three?
- When the minute hands move 360 degrees, the hour hands move by 30 degrees
- Since at 3, the minute hand has moved by 90 degrees which is 1/4 of its total journey
- 1/4 of 30 = 7.5
- By the time, the minute hand gets to 3, the hour hand has moved by 7.5 degrees (this is the angle between the hour and minute hand).
What is the first time after 3PM when the hour and minute hands of clock are exactly on top of each other?
- As the minute hands whips 60 minutes around the entire face, the hour hands only moves by 5 increments (5 - 1 minute marks)
- At all times, the proportion of the full 60 mins traversed by the minute hand equals the proportion of the five increments of one minute traversed by the hour hand.
- M/60 = (M-Hour After)/ 5
- Our example:
- M/60 = (M-15)/5
- M = 180/11 = 16(4/11) Minutes
- Answer: 3 hours, 16 minutes and 4/11 of the minute
100 Light bulbs. 100 Brokers. Initial state of bulbs = Off.
1st Broker flips the switch for every single bulb (so it turns on). 2nd Broker flips the switch for every 2nd bulb (2 bulb turns off), and 3rd broker flips the switch for every 3rd bulb. This continues untill all 100 brokers have passed. **What is the final state of bulb number 64? **
**How many light bulbs are illuminated after the 100th person has passed - which light bulbs are they? **
- Bulb will be on
- Factors of 64
- 1 ,2 ,4, 8, 16, 32, 64
- Since there are odd numbers of factors and the bulb was initially off. The bulb #64 would be on
- The only light bulbs illuminated at the conclusions are those with a number that has an odd numbers of factors
- However factors go in pair:
- ex: 32, (1, 32), (2, 16), ..
- Except: one pair of factors are identical, then the original number must be a “perfect square”
- There are exactly 10 perfect squares (1 to 100)
First person to call out “50” wins.
Rule 1: The player who starts must call an integer between 1 and 10
Rule 2: A new called out must exceed the most recent number by at least one and no more than 10.
What is my strategy?
- I want to go first
- I want to call 39
- I have to call 28
- I have to call 17
- I should call 6
- Winning series:
- 6, 17, 28, 39, 50 (I will not loose)
Goal is to open a safe. There are three combination numbers. Dial has number from 0 to 40.
Without knowing the combination numbers - what is the max number of trials required to open the safe?
- 40^2 = 1600
- Why? Because we don’t need to know the last combination - we can keep rotating till we actually get the result we need.
10,000 Ants are dropped onto a ruler that is 100 centimeters Long. Each ant walks at a steady pace of 1 centimeter per second. If each ant collide with each other (head on) then each turn the other way. There can be more than one collison. How long must you wait to be sure that all the ants have walked off the ruler?
It won’t take more than 100 seconds. Answer is the same if ants had the perfect vision. Key: if two ants collide immediately about face and continue, then each member of any colliding pair effectively exchanges its exit route with the other. This is the same as one of the colliding pair crawled over the other and they both kept going without a pause.
One Lily Pad. Surface area doubles every day. It will take 30 dayy to cover the surface of the pond. If instead of one lily pad, we start with 8 lily pads, how many days will it take for the surface area of the pond to become crowded?
- 20, 21, 22, 23…
- 8 Lilly pad is the same thing as one lily pad on the thrid day and after that pace of increase is the same.
- Hence it would take about 27 days to for the surface of the pond to become covered.
