General Flashcards
Two glass jugs, one with alcohol and one with water. First a bit of one is poured into the other, than the same amount is poured back. What are the end concentrations?
- after the first pour the dilution is A/W+A
- after the second pour the dilution is W/W+A
- concentration is opposite
- compare to blocks.
“You are a bug sitting in one corner of a cubic room. You wish to walk (no flying) to the extreme opposite corner (the one farthest from you). Describe the shortest path that you can walk. Be sure to mention direction, length, and so on.”
- Geodesic
- 5^1/2
Fischer and Myron just stepped side-by-side onto a moving escalator. They are climbing up the stairs, and counting steps as they climb. Myron is climbing more quickly than Fischer. Myron climbs three steps in the time it takes Fischer to climb only two steps. Neither of them skips any steps. Myron steps offat the top, having counted 25 steps. He waits at the top for the slower Fischer, who steps offhaving counted only 20 steps. How many steps are showing on the escalator at any instant?
- D = X + 25
- D = (X*k) + 20
- F is at 16 2/3 when M finishes, so will take 20/(16 2/3) = 1.2 longer -> k = 1.2
- 1.2x + 20 = x + 25
0.2x = 5 -> x = 25
D = 50
There are two bells. One rings five times per minute, and the other rings four times per minute. If they start at the same time, how long will it be until they next ring together?
- 1 rings every 12 seconds
- 2 rings every 15 seconds
- the next number that can be divided by 12 & 15 is 60, so at one minute.
What is the sum of the integers from 1 to 100?
- Gauss summation
- 50 * 101 = 5050
- 0.5n * n+1
An old style analog clock falls offthe wall and the face breaks into three pieces. The numbers on each piece add to the same total. Describe the pieces.
- clock face has numbers 1 - 12, which summed are 78
- each piece needs to sum to 26
- therefore; 11+12+1+2, 10+9+3+4, 5+6+7+8
You are given a set of scales and 12 marbles. The 12 marbles appear to be identical. In fact, 11 of them are identical, and one is of a different weight. Identify it in 3 weighings
- Split marbles in groups 1,2,3,4 ; 5,6,7,8 ; 9,10,11,12
- Compare the first two groups.
2A. If group 1 is heavy, compare 1H,5L,6L with 2H,7L,8L.
2AA. If 1H,5L,6L is heavy then either 1H or 7L or 8L. Hence compare 7L & 8L.
2AB. If 1H,5L,6L is light then either 2H or 5L or 6L. Hence compare 5L & 6L.
2AC. Either 3H or 4H
2B. If group 2 is heavy, compare 5H,1L,2L with 6H,3L,4L as above
2C if plates are equal then 9H/L, 10H/L, 11H/L, 12H/L. Compare 1,2,3 with 9,10, 11 to determine if H or L
2CA. If L,/H compare 9 & 10 to determine 9, 10 or 11 L/H
2CB if equal, compare 1 & 12H/L to determine H/L
Suppose I inscribe a circle within a square so that the circle just touches the four sides of the square. Suppose there is exactly enough room to fit a rectangle of dimensions 5 × 10 into one corner of the square so that the rectangle just touches the circle. What’s the length of the square
1 realize you can draw triangle R, (R-5),(R-10)
2 per Pythagoras, R^2 = (R-5)^2+(R-10)^2, so
R^2 = R^2 -10R+25+R^2-20R+100 so
R^2-30R+125 = 0.
3. -B+-(B^2-4AC))^1/2 /2A = 30+-(900-500)^1/2 / 2 -> 30 +-20 /2 ~> 25 v 5
4. Square length is 25*2 = 50
A mythical city contains 100,000 married couples but no children. Each family wishes to “continue the male line,” but they do not wish to overpopulate. So, each family has one baby per annum until the arrival of the first boy.
Let p(t) be the percentage of children that are male at the end of year t. How is this percentage expected to evolve through time?
Constant at 50%: 50.000-50.000, 25.000-25.000, 12.500-12.500
Picture a 10 × 10 × 10 “macro-cube” floating in mid-air. The macro-cube is composed of 1×1×1 “micro-cubes,” all glued together. Weather damage causes the outermost layer of micro-cubes to become loose. This outermost layer falls to the ground. How many micro-cubes are on the ground?
- 2 x 10 x 10
- 4 x 8
- 4 x 8 x 8
- 200+ 32 + 256 = 488
- or 10^3-8^3 (=10^3-2^9)
There are two cities, A and B, 1,000 miles apart. You have 3,000 apples at City A, and you want to deliver as many as possible of them to City B. The only delivery method available is a truck. There are, however, two problems. The truck can hold at most only 1,000 apples, and if there are any apples at all in the truck, the hungry dishonest driver will steal and eat one apple for every mile he drives. What is the maximum number of apples you can deliver from City A to City B? Note that you are welcome to stop part way, dump offsome apples, and then come back and pick them up later.
- dump in same place
- never drive with less than thousand
- single dump for each drive
- In general, starting with N × 1, 000 apples, we will carry them (spread over N trips) until we can dump (N − 1) × 1, 000 apples. If we dump the apples at the k-mile marker, it must be that N ×(1, 000−k)=(N −1)×1, 000. Solving for k yields k = 1,000 N. So, to go from 3,000 apples down to 2,000 apples we dump them at the 1,000 3 -mile marker. Assuming we cannot slice apples up to get a continuous solution, let us dump them at the 333-mile marker. We will have T = 2, 001 apples dumped there. Ignore one apple. Using the same argument again, we will drive the 2,000 apples in two trips, dumping them in two batches after another 1,000 N = 1,000 2 = 500 miles. Then we will have 1,000 apples at the 833-mile marker. We can then drive them on the remaining 167 miles in one trip, delivering 833 apples.13
How many degrees (if any) are there in the angle between the hour and minute hands of a clock when the time is a quarter past three?
- minute hand: 360/60 = 6 degrees per minute. So @ 90
- hour hand = 360/12*60 = 0.5 degrees per minute + hour (90). So @ 90+7.5
-7.5
What is the first time after 3pm when the hour and minute hands of a clock are exactly on top of each other?
Method 1
- minute hand: 360/60 = 6d per minute
- hour hand = 0.5d per minute
6x = 90 + 0.5x -> 90=5.5x -> x = 16 20/55 min
Method 2
- happens each hour, first @ 360/11 = 32 8/11 degree; 13.06 8/66 minute. third @ 32 8/11*3 = 96 + 2 2/11 minute ~ 15.16 20 sec
There are 100 light bulbs lined up in a row in a long room. Each bulb is numbered consecutively from 1 to 100. Each stockbroker is numbered consecutively from 1 to 100. Broker number 1 enters the room, switches on every bulb, and exits. Broker number 2 enters and flips the switch on every second bulb.
What is the final state of bulb number 64? Is it illuminated or dark?
- everything is touched as often as it can be divided
- 64 is divisible by 1, 2, 4, 8, 16, 32, 64 (7 numbers) so it is on
There are 100 light bulbs lined up in a row in a long room. Each bulb is numbered consecutively from 1 to 100. Each stockbroker is numbered consecutively from 1 to 100. Broker number 1 enters the room, switches on every bulb, and exits. Broker number 2 enters and flips the switch on every second bulb.
How many of the light bulbs are illuminated after the 100th person has passed through the room, and which light bulbs are they?
- everything that is divisible by an even amount of numbers is off, others are on
- only squares are divisible by odd amount
- so only squares are on
