ERJ Brainteaser: July

Fabio's favourite number is 4. He rolls a dice four times to see if he can get at least one 4. Next he rolls two dice 24 times to see if he can get at least one double-4. Which is more likely to work?

Answer: Many readers went for the obvious answer that the probability was the same for both, ie 2/3. Indeed, it was only Michele Girardi, Scame  Mastaf Spa, Suisio, Italy; and Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; who identified that the first approach is more likely to work .  

As Michele Girardi explained: Let's calculate the probability to fail: 
- in the first case it's (5/6)^4 = 48.2% 
- in the second case it's (35/36)^24=50.9%

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Answer: This missing-number style of question has proved quite a challenge, even for many of our top Brainiacs, over the last couple of months. But not so this week for Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; and Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands, who both came up the correct answer 85. 

Solution: The number in the top row is the total from adding the square of each the numbers in the bottom row: (2 x 2) + (9 x 9) = 85.
 

Bonus questions: How many rectangles?

If you were to construct a 7 x 7 checkered square (i.e., a 7 x 7 chess board), how many rectangles would there be in total? You need to include squares too because a square is a special kind of rectangle.

Answer: This week’s BQ was a genuine toughie, as evidenced by the lack of correct replies.  Here goes with the solution:

Length of rectangle and number of Possibilities
7 units 1
6 units 2
5 units 3
... ...
1 unit 7
So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Hence, number of rectangles = 28 x 28 = 784.

Again, many thanks to everyone who had a try – they won’t always be this difficult. 
 

Last year, endurance athlete Amy won the annual women’s race up the 1576 steps of the Empire State Building in New York in a record time of 11 minutes 57 seconds. Approximately how many steps did she climb per minute?

Answer: Our high-flyers this week – in order-of-reply with the correct answer 132 steps per minute – were: John Bowen, consultant, Bromsgrove, UK: Michele Girardi, Scame  Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Daniel Willrich, redakteur,  AutoRäderReifen-Gummibereifung, Hannover, Germany; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada: Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Vivian Zhou, senior business development & market intelligence analyst. BU RE PLT APAC, Continental Tires (Shanghai) Co. Ltd, Shanghai, China. Well done to all, and everyone else who had a go.

Bonus question (see below)

The key was to translate the letters into their numerical position in the alphabet, so that A = 1, B – 2, C = 3 etc. Then it was about realising that the number in each bottom row was the result of subtracting the number in each second row from the number in each top row. Only one Brainiac managed to battle through the challenges here, including our typo last week. Extremely well done to Andrew Knox for answering ‘I’. (18 + 24 + 4 + 8) – (16 + 7 + 12) = 54 -35 = 19 / AI.
 

Missing letter (corrected and carried over from last week)

With apologies, there was a blip with the question, as posted this week. We will correct and run as a Bonus question next week.

The date of publication of the archived article was 23 Jan 1932: seems the cartoon was about government efforts to encourage spending amid or in the wake of the Great Depression.

Bonus question

Try to guess the year-of-publication of the following snippet from the ERJ archives, which date back to 1884. 

At a chemical plant, a tank is filled with two liquids through tap A and tap B on top and emptied via valve at the bottom. Tap A on its own fills the tank in 15 minutes and tap B in 6 minutes. When opened, the valve can drain the tank in 30 minutes with both taps closed off. How long will the tank take to fill if the two taps and the valve are fully opened at the same time?

Answer: A nice, steady start to the month. Well done in order of correct-reply to: France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; Vivian Zhou, senior business development & market intelligence analyst. BU RE PLT APAC, Continental Tires (Shanghai) Co. Ltd, Shanghai, China; Daniel Willrich, redakteur, AutoRäderReifen-Gummibereifung, Hannover, Germany; David Mann, key account manager, SPC Rubber Compounding, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria.

Solution

Andrew Knox
Answer: 5 minutes.
Just assume the tank has a capacity of 600 litres (for ease of mental arithmatic using common denominators).
Tap A fills the tank in 15 minutes so rate of entry is 600/15 = 40 litres/minute.
Tap B fills at a rate of 600/6 = 100 litres/minute. So the tank fills at a constant rate of 140 litres/minute with both taps fully open.
The valve empties in 30 minutes. This an average rate of 600/30 = 20 litres/minute. 
The tank drains faster to start with, due to a higher pressure at the exit valve. The variable rate at which the tank empties whilst filling however is the exact reverse of the variable rate when the tank emptied. 
The net filling rate is therefore still (40 + 100) - 20 = 120 litres/minute, so the tank fills in exactly 600/120 = 5 minutes.
The tank however is more than half full after 2 1/2 minutes!

Daniel Willrich
it takes 5 minutes to fill the tank. Each minute 1/15 of the tank is filled by tap A plus 1/6 of the tank is filled by B, while 1/30 of the tank is emptied through the valve. This means 7/30 is going in with 1/30 going out, which is why 6/30 or 1/5 of the tank is filled per minute. 

Jose Padron
First, we need to know the flow of each tap and valve
Flow = F Volume = V Time = t
Flow is given by F = V/t; from this, t = V/F 
In the other side: Total flow = flow tap 1 + flow tap2 – flow drain valve
As volume is constant, we have
Total flow = F = V/t1 + V/t2 – V/t3
F = V/15 + V/6 – V/30
Solving t = V/F we have: t = V/V(1/15+1/6-1/30)
Cancelling V; we have: t = 1/(1/15+1/6+1/30) = 5
 

Bonus question

Very well done to France Veillette and Stephan Paischer who were spot on with 1911. Special mentions also to Andrew Knox 1912-13, and not too far out: Paul Knutson,1905; and David Mann, 1901. The article was actually published in an issue dated 4 Nov 1911.

Try to guess the year-of-publication of the following snippet from the ERJ archives, which date back to 1884.