ERJ Brainteaser - January 2020

After checking carefully through all the rule books, we could find nothing to prevent readers combining their brainpower to answer our weekly teasers. So, this month’s top award goes – for their use of initiative – to Chrystelle Tandonnet and Indrayush De of KSB SAS Pompes, as well as to Andrew Knox of Rubbond International – for his excellent answering, especially of tricky Q4.

Congratulations to all three: the new joint holders of our prestigious Brainiac of the Month title.
 

A digital clock on Simon's glass coffee table shows the time as 13:08. Simon notices that the mirror image of the time on the glass is exactly the same as the real time.

How many times during 24 hours the clock and its mirror image show the same time?

Answer: This one proved quite a challenge for even our top Brainics with just two correct answers – 96 times per 24 hours – among the replies that came in. So extra well done to Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria and Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands;

Andrew Knox explains how it can be done.

If Simon's clock shows 13:08, then it is a 24 hour clock.
Digits from 0-9 which show the same image when reflected from a table top are 0,1,3 & 8.
Of the 4 position on the display, the first position can therefore only be 0 & 1, the second position can be 0,1,3 & 8.
So for the hours, the possibilities are:
0  0
0  1
0  3
0  8
1  0
1  1
1  3
1  8
So, 8 times in 24 hours the hours reflect correctly. This is 2 x 4 =8 possibilities.
For the minutes, the 3rd position can be 0, 1 or 3, the 4th position can be 0,1,3 & 8.
So, for the minutes, there are 3 x 4 = 12 possibilities.
For each correct hour, there are 12 possibilities for the minutes to reflect correctly.
So, there are 8 x 12 possibilities in 24 hours, which is 96.

Two 2s can be combined in many ways to express different numbers. Here are some!

2-2=0

2/2 = 1
2 + 2 = 2.2

(2^2)! = 24 (4! means 4x3x2x1) (2^2 is 2 to the power of 2)

Can you write an expression that has the value of exactly 5, using:

• Two, and only two, 2s, and
• Any mathematical symbols or operations?

Answer: This teaser was a test of both maths and keyboard skills:

Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands;
The square root of (.2 to the power of minus 2), i.e. the square root of (1/.2) x (1/.2), or the square root of 1 divided by .04, or the square root of 25.

John Bowen, consultant, Bromsgrove, UK:
Sq Rt of [0.2 to the power of  -2] =5 / ie  Sq Rt [1/0.04]

Michele Girardi, Scame  Mastaf Spa, Suisio, Italy;
5= sqroot(25)=sqroot(100/4)
5 = Sqroot( inverse.log(2) / inverse.sqroot(2) 

Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA
.2^(-2^0)

Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; 

Chrystelle Tandonnet and Indrayush De, application & technology development, KSB SAS Pompes. Gradignan, France: 
SQR(.2^-2)
? (0.2^-2)

And finally this extra one given by a “mathematician passing nearby” Michele Girardi’s desk: 5 = (2-i)*(2+i).

Well done to all above and everyone else who had a go.

An equilateral triangle and regular hexagon have perimeters of the same length.

What is the ratio of the area of the triangle to the area of the hexagon?

Answer: The ratio of triangle area to the hexagon area is 2/3, as answered correctly, in order of reply, by: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Michele Girardi, Scame  Mastaf Spa, Suisio, Italy; Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; Fariha Rashid, marketing analyst, Kraton Polymers LLC, Houston, Texas, USA; David Mann, key account manager, SPC Rubber Compounding, UK; Chrystelle Tandonnet and Indrayush De, application & technology development, KSB SAS Pompes. Gradignan, France; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA.

Many thanks for some great replies, but we will let Jose Padron show how it could be done, both graphically and mathematically:

Fractioning the hexagon in six equal equilateral triangles and fractioning, as well, the whole triangle into four equal equilateral triangles:

Superposing the triangle over the hexagon

And a mathematical model

There are 9 counters in a bag.

There is an even number on 3 of the counters.

There is an odd number on 6 of the counters.

Three counters are going to be taken at random from the bag.

The numbers on the counters will be added together to give the total.

Find the probability that the total is an odd number.

Answer: A Happy New Year to all our readers. The answer to our first teaser of 2020 is 45%, or as Andrew Knox more precisely calculated 45.2382%. Well done, in order of reply, to: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Chrystelle Tandonnet, materials & process expert, application & technology development,KSB SAS Pompes. Gradignan, France; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; David Mann, Polymer Business Development, France;

Many thanks for all the neatly worked out solutions, such as the following from David Mann:

There are 8 ways of drawing the three counters, giving the probabilities of each draw and the combined draws in the table below. So the probability of an odd total is 0.45: