Problems

Some brain teasers challenges I got, my take on them, and solution! 😃

Building: Tape & Watch

Get exact height of the Empire State building using a tape measure and a watch

The building and the floor build a right angle. You and the building also form a smaller and contained right angle. So you have a smaller and contained (you) right triangle in a larger (building) right triangle.

Use the measure tape to measure your shadow and also the building shadow at the same time of the day. Early in the morning or late close to the evening are very long shadows and your tape measure might be not very long. It is best to measure sometime around noon.

\(\color{purple}{\textbf{Solution}}\). If \(h\) denotes the heights, \(s\) the shadows, and \(\alpha\) the tangent of the angles, then: \(\alpha=h_b/s_b=h_p/s_p\). Use the information of you (person) and your shadow to compute \(\alpha\) and then using that and the measured building shadow \(s_b\), you can get the height of the building \(h_s\).

Social: Dead vs Alive

When will we see more profiles of dead than alive people on Facebook?

Some statistics to bear in mind: Active users (by end 2018): 2.27 billions, and new active users (per year): 0.27 billions (2018 vs 2017). So a rate of 13.5% yearly (I will later round it to 15% and assume it to be exponential). World population: 7.7 billions. World death rate: 1.07% per year.

Assumptions to reach a solution: 1) The yearly death rate has been constant in the last 10 years (~ when FB started). Note: It is not a very restrictive assumption; 2) The population of FB users has the same mortality rate as the overall world population. Note: Not very realistic. They are “rich” people, which have a lower death rate; 3) The yearly growth rate of new active users of FB has decreased exponentially over the years. Note: Quite realistic. At first the growth rates were very high and now they are very low.

According to some research there will be more profiles of dead people than live people by the end of the century (in 70 years). Here is some simple math to sustain that claim:

Active users 10 years ago in 2008? We take the current users and discount them exponentially at the 15% rate: \(2.27e^{(-0.15*10)}=0.5\) billion users.

Dead profiles since 2008? Users have grown exponentially and also died linearly 1% every year, so we have: \(0.5e^{(0.15*10)}(1+0.01)^{10}=0.5*2.27*1.10=1.24\) billion dead profiles in 2018.

\(\color{purple}{\textbf{Solution}}\): So for the current 1.24 billion dead profiles and 2.27 billion active users, we have that \(y\) below gives the number of years it takes us have the same amount of dead and live profiles:

\(2.27e^{(0.15*y)}=1.24e^{(0.15*y)}(1+0.01)^{y}\)

Note that dead profiles is given by the evolution of profiles (15%) and the evolution of the number of people or death rate (1%). Then doing some algebra we have that:

\(2.27/1.24=(1+0.01)^{y}\) => \(1.83=(1.01)^{y}\) => \(log_{1.01}1.83=y\) => So the solution is \(y=61\).

You get the available result of 70 years using a death rate of 0.8%. Also note that probably the rate of growth of new FB users is not exponential with 15% (that gives an incorrect number for users in 2008).

Roman Numbers

Without touching the board or adding any sticks, how can you make XI + I = X right?

The equation reads as “11 + 1 = 10” which is false. And we need to fix it without adding any sticks or even touching the board. That leaves us with few options on what can we do.

\(\color{purple}{\textbf{Solution}}\): Tell the author to turn his back to the board, stick his head between his legs, and look at the board. It now reads \(X = I + IX\), which is “10 = 1 + 9”.

Clock: Time as Angles

If the time in the clock is 12:15, what is the angle between the hour and the minute hand?

We know that at 12:15 the angle is slightly less than ¼ (a bit less than 90). Why?: because the hour hand is a little past 12, since it needs to move from “12” to “1” over the course of an entire hour.

A clock, which is a circle, is 360 degrees. Each hour on the clock represents 30 degrees (360 degree/12 hours). So the hour hand has been moved in 7.5 degrees: 30 degrees per hour times ¼ hour move in the minute hand, representing the 15' that have lapsed.

\(\color{purple}{\textbf{Solution}}\): The angle is 82.5 degrees (90-7.5) since the hour hand was pushed 7.5 degrees in 15' lapsed.

Car Speed for Average

Speed of travel back 60 miles (done at 30 mph) to average 60 mph over the entire trip?

In order for you to average 60 mph over the entire trip (120 miles) you would have to travel for 2 hours (120 miles / 60 mph). The first trip already took 2 hours (60 miles / 30 mph).

\(\color{purple}{\textbf{Solution}}\): Since you already were driving for 2 hours it is impossible for you to average 60 mph for the entire trip. There is no speed, no matter how fast, to achieve that.

Gallon Jugs to Measure

You have a 5-gallon jug and a 3-gallon jug. You must obtain exactly 4 gallons of water. How?

First you try to get 2-gallon in each jug. For this, fill the 5-gallon jug and then pour it into the 3-gallon jug. The 5-gallon jug has 2-gallons of water left.

Now throw away the water in the 3-gallon jug. Pour the remaining 2-gallons from the 5-gallon jug to into the 3-gallon jug. So now the 3-gallon jug has just 2-gallons of water.

Now throw away the water in the 5-gallon jug. Fill it again. Use it to fill the 3-gallon jug to the top. Then you know the 5-gallon jug will have remaining 4-gallons of water.

\(\color{purple}{\textbf{Solution}}\): Get 2-gallons in each jug, empty the 5-gallons jug and use it to fill the remaining 1-gallon missing in the 3-gallon jug.

Weight of Lighter Marble

Minimal use of a scale to determine which one of the 9 marbles is the lighter one?

1st weighing: you would weigh three marbles on each side (1,2,3 vs 4,5,6), leaving three off (7,8,9).

2nd weighing: If one side of the scale is lighter, you are left with three marbles (say 1,2,3). Then you would place one marble on each side of scale (1 vs 2), and leave one off (3). You can now conclude if 1 or 2 was the lighter one, and if they weight the same, then the lighter one was marble 3.

\(\color{purple}{\textbf{Solution}}\): You need to use the scale only twice. Once to figure out the marbles in groups of three and the second weighting to do weight single marbles.

Bridge Crossing

How can 4 people cross it in 17 minutes, two at the time, at night, only one flashlight?

Four people are on one side of a bridge at night and would like to cross it. Only two can cross at a time. In order to cross, a flashlight must be used. There is only one flashlight. People will cross it at the speed of the slowest person. How can all cross in 17 minutes? Person A takes 1 minute, Person B takes 2 minutes, Person C takes 5 minutes and Person D takes 10 minutes.

• A & B cross (2 minutes), A comes back with flashlight (1 minute).

• C & D across (10 minutes), B (from first crossing) comes back with flashlight (2 minutes).

• A & B across (2 minutes).

Total Time = 17 minutes

\(\color{purple}{\textbf{Solution}}\): Make the two fastest cross first, and the fastest of those comes back. Then cross the two slowest ones, and bring back the remaining of the fastest ones. Then the two fastest cross over again.

Weighting Unkown Coins

1 out of 10 machines produces coins a gram lighter. Which one is? You can do one weighting

• Note that there is only one machine that is defective.

• You have to weigh all these coins from different machines all together.

• Even if each machine produces a different number of coins this will not be a problem. As long as you can see how many coins each machine was spitting.

• Without loss of generality, assume that Machine 1 produces 1 coin, Machine 2 produces 2 coins, Machine 3 produces 3 coins, and Machine 4 produces 4 coins.

\(\color{purple}{\textbf{Solution}}\): Weigh all the coins together against the theoretical weight, which is 10 in this case. Assume, without loss of generality, that you are 4 grams short. Then, since only one machine is defective, it has to be the one that spat 4 coins, that is, Machine 4.

Extra from Dinner Tip

Where did the extra dollar came from?

You and your friend go out to dinner together and the bill is 25. You and your friend each pay 15. The waiter keeps 3 as a tip and hands back 1 to each of you. So you and your friend paid 14 each for the meal, for a total of 28. The waiter has 3, and that makes 31.

So the confusing part of the statement is that we handed in 30, but apparently the waiter ended up having 31 (28+3). That is the “extra dollar”.

The 30 means 12.5 (meal) + 1.5 (tip) = 14, per each friend, or a total of 28, and we expect back 2 dollars. The 31 means a 28 that already includes a tip, plus a 3 of double counting the tip. The difference between both numbers, 30 and 31, is the expected 2 back plus the 3 on tip. That is a net error of 1.

\(\color{purple}{\textbf{Solution}}\): The extra dollar comes from you forgetting that for each friend you get cancelled the 1 they receive as change with part of what they gave on tips (the 0,5 part of the 1,5 for each friend).

Switches and Light Bulbs

How can you know which switch corresponds to bulbs located upstairs?

You are standing at three light switches at the bottom of the stairs to the attic. Each one corresponds to one of the three lights in the attic, but you cannot see the lights from where you stand. You can turn the switches on and off and leave them in any position. How can you identify which switch corresponds to which light bulb if you are only allowed one trip upstairs?

Turn on the first two switches (i.e., 1 and 2 starting from the left) and leave them on for 5 minutes. After 5 minutes, turn off the second switch, leaving the first switch on. Now go upstairs to the attic.

The light that is on is connected to the first switch. A light that is off but has a bulb that is still warm to the touch is connected to the second switch. The light that is both off and cold to the touch is connected to the third switch, which was never turned on.

\(\color{purple}{\textbf{Solution}}\): Turn on two switches, leave them on for a while, turn one off. Go upstairs. One bulb is on, another one is warm, the other one is cold. Voliá. They are identified.

Burning Ropes for Time

How do you use two different ropes to measure 45 minutes of time?

You have two ropes, each of which takes 1 hour to burn. But either rope has different densities at different points, so there’s no guarantee of consistency in the time it takes different sections within the rope to burn. How do you use these two ropes to measure 45 minutes?

Note: For a rope that takes \(X\) minutes to burn, if you light both ends of the rope simultaneously, it will take \(X/2\) minutes to burn.

Light both ends of the 1st rope, and light one end of the 2nd rope. Then 30 minutes later, the 1st rope will get completely burned, while the 2nd rope now becomes a 30-min rope.

At that moment, we can light the other end of the 2nd rope at the other end (with the other end still burning), and when it is burned out, the total time is exactly 45 minutes.

It is key to note that one of the extremes of the 2nd rope has been already burning for a while, and that it will keep on burning while I light the other end. This guarrantees the 15 extra minutes I need to be sure I will get the total of 45 minutes that the task is asking me.

\(\color{purple}{\textbf{Solution}}\): Light one rope by both ends, and you know it will burn in 30 minutes, while at that same time light only one end of the other rope. When the first one is completely burnt, light the other end.