ERJ Brainteaser: February

Question 4: Colour conundrum

Art teacher Sam is organising some sets of coloured drawing pencils, when two accidently drop down behind his desk. He saw a blue one falling and knows that the other pencil must be either also be blue or green.

If after reaching down, he pulls up a blue pencil, what is the probability that the other one is also blue?

Answer: Okay, there were alternative interpretations - many readers answering 50% - however, officially the probability is 2 / 3, based on the logic:

Blue fell first followed by blue

Blue fell first followed by green

Green fell first followed by blue

Closest to the mark was this impressive solution from Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy:

To make a precise calculation you need to know the number of colors and the number of kits.

The combinations of pencils' falls that match the observation are blue/blue, blue/green, green/blue.

Blue/green and green/blue have the same probability, blue/blue is slightly lower, but we assume it's the same.

So, the probability that the fallen pencils are blue/blue is 1/3 = 33.3%

For an example of precise calculation let's consider 12 colours and 10 students.

The probabilities are in absolute

blue/blue    10/120*9/119 = 0.0063

blue/green and green/blue   10/120*10/119 = 0.0070

for the 2 pencils

blue+blue    63/(63+70+70) = 0.31 = 31%.

Well done to everyone who had a go - comments about the solution welcome.

New teaser on Monday

The combinations of pencils' falls that match the observation are

blue/blue, blue/green, green/blue   .

Blue/green and green/blue have the same probability, blue/blue is slightly

lower but we assume it's the same.

So the probability that the fallen pencils are blue/blue is 1/3 = 33.3%

For an example of precise calculation let's consider 12 colours and 10 students .

The probabilities are in absolute

blue/blue  10/120*9/119 = 0.0063

blue/green and green/blue 10/120*10/119 = 0.0070

for the 2 pencils blue+blue 63/(63+70+70) = 0.31 = 31%.

Well done to everyone who had a go - comments about the solution welcome...

New teaser on Monday.

We tally correct responses by country each month, now with 10 points per correct answer and extra points for awards (20) and special mentions (10). The final ranking for February is: Italy (70) / UK (60) / Spain (40) / Austria, Netherlands, USA (30) / Canada, (20) / Belgium, Turkey (10).

Question 3: Missing link

What links the following numbers?

1889, 300 (324), 125x125, 10,100

Answer: From the reply locations, it seems that this question was more of a challenge for readers outside of Europe – will try to balance this up in a future teaser.  Nevertheless, it took sharp eye and lateral thinking to work out the answer: The Eiffel Tower was built in 1889, is 300m high or 324m with the radio mast, has a base measuring 125m x 125, and weighs 10,100 tonnes. Bien joué: Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Andrew Knox, Rubbond International, The Netherlands; Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy; David Mann, key account manager, SPC Rubber Compounding, UK.

Question 2: Odd numbers

11 x 11 = 4

21 x 21 = 9

31 x 31 = ?

Answer: Turns out that these numbers were even odder than expected, with more than one way to get to the answer 16 (see solutions below. Very well done to: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; David Mann, key account manager, SPC Rubber Compounding, UK; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Michel Meuwissen, M+M Direct bvba, sourcing and consulting company, Zutendaal, Belgium; Andrew Knox, Rubbond International, The Netherlands; Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy; Jose Padron, laboratory analyst, Waterville TG Inc., Waterville, Québec, Canada; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc., USA; Mehmet Koral, Erhardt-Leimer representative for Turkey, managing director, C&C Endüstriyel Dan??manl?k, E?itim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey; Mark Bussy, M. Rothschild & Co., USA; and everyone else who had a go.

Solutions

John Bowen

Each answer is the product of the sum of the insividual digitd:

11 x 11 = [1+1] x [1 + 1] = 2

21 x 21 = [1+2] x [1+2] = 9

31 x 31 = [3+1] x [3+1] = 16 

Amparo Botella

1 x 11= 4 (1+1 x 1+1) = 2x2=4

21 x 21= 9 (2+1 x 2+1)= 3x3=9

31 x 31= 16 (3+1 x 3+1) = 4x4= 16

Now I saw the [clue]:

 11x11 = 121 = 1+2+1= 4

21x21=441=4+4+1=9

31x31=961=9+6+1=16

David Mann

One solution is to sum the digits, then:

2 x 2 = 4

3 x 3 = 9

31 x 31 => 4 x 4 = 16 

Stephan Paischer

Replacing the figures with the digit sum creates a valid formula.

11 x 11 = 4       2 x 2 = 4   digit sum of 11 = 2

21 x 21 = 9       3 x 3 = 9   digit sum of 21 = 3

31 x 31 = 16     4 x 4 = 16   digit sum of 31 = 4 

Andrew Knox

Add individual digits before performing the multiplication I.e. 11 x 11  becomes 2 x 2 = 4, 31 x 31 becomes 4 x 4 = 16

Clue: 11 x 11 = 121…

Your clue today suggests that your solution requires conventional multiplication of the numbers first, then addition of the individual digits in the product, i.e. 11 x 11 = 121 and 1 + 2 + 1 = 4.

As your puzzle does not go further than 31 x 31 = ?, by chance it does not make a difference if you add the individual digits first and then multply, or multiply first and then add the individual digits.

In both cases the answer is 16 (or should that be 5?):

31 x 31 = 961, 9 + 6 + 1 = 16, or  (3 + 1) x (3 + 1) = 4 x 4 = 16

However, beyond 31 x 31 it does make a difference: 41 x 41 = 1681, and 1 + 6 + 8 + 1 = 16, whereas 5 x 5 = 25.

Personally I find adding the individual digits first, then multiplying more satisfactory, and easier to do ithe calculation in one's head.

Given your clue however, the correct answer to 41 x 41 = ? would be 16 too (but you didn't ask that).

Michele Girardi

The answer is 16, as  we can see from this sequence :

11x11 = 121     1+2+1 = 4

21x21 = 441     4+4+1 = 9

31x31 = 961     9+6+1 = 16

Jose Padron

First option:

Let us try this: 16, as described below.

Addition of individual digits and then multiplication of factors:

(1+1) * (1+1) = 4 = 2²

(2 + 1) * (2 + 1) = 9 = 3²

(3 + 1) * (3 + 1) = 16 = 4²

Second option:

In a contrary sense;

Multiplication of factors and then addition of individual digits:

11 * 11 = 121 à  1 + 2 +1 = 4

21 * 21 = 441 à  4 + 4 +1 = 9

31 * 31 = 961 à  9 + 6 +1 = 16

Surely, there are another possibilities, that is the reason they call them odd numbers, isn’t it? 

Rohit Kalé

The result is derived from multiplying 31*31 = 961 and then adding the digits of the answer.

11           X             11           =             121         1+2+1=4

21           X             21           =             441         4+4+1=9

31           X             31           =             961         9+6+1=16

Mehmet Koral

Given odd numbers are as follows;

11 x 11 = 4    (1+1)x(1+1) = 2x2= 4

 21 x 21 = 9       ( 2+1)x( 2+1) = 3x3= 9   

31 x 31 = ?

The rule to be applied; The result is  [ The  square  of (the  total of the given digits in each Odd number) ]

Hence, for the given odd number 31 x 31 =?

(3+1) x  (3+1) = (4) x (4)=16 is the result.

Question 1: Enigmatic (4)

Work out the missing number:

France (_), Denmark (4), Argentina (6) Burkina Faso (8).

Answer: Probably among our toughest ever teasers and so impressive that Michele Girardi, quality manager, Scame Mastaf Spa, Suisio, Italy and Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain were able to figure this out before we issues clues Capital and AEIOU. The enigmatic key was that the figure in brackets indicated the number of vowels in the names of the capital cities of those countries. So, France (2) for Paris. starts the series Denmark, Copenhagen (4), Argentina, Buenos Aires (6), Burkina Faso, Ouagadougou (8). Very well done also to: Andrew Knox, Rubbond International, The Netherlands; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc., USA; David Mann, key account manager, SPC Rubber Compounding, UK; Jose Padron, laboratory analyst, Waterville TG Inc., Waterville, Québec, Canada; and everyone else who had a go.