*List of Brainteaser questions over the past couple of months, along with the answers and readers supplying correct replies. The number **of correct answers, as well as speed and quality of reply will help decide our Brainiac of the Month. *

**MARCH 2019**

Question 2: **Teacher teaser**

In introducing herself. a new maths teacher tells her class that she has three children, before asking: if at least one of the children is a boy, what is the probability that she has three sons?

**Answer**: There were several different interpretations of this question, but the actual answer is 1/7 (or 14.3%), as all possible ways of the teacher having the three children must be considered – (ie BBB, BBG, BGB, GBB, GBG, GGB, BGG, but not GGG).

Extra well done to Michele Girardi, Scame Mastaf Spa, Suisio, Italy, who was the only one to get the correct answer, neatly explaining: There are 8 possible combinations of children, only one has no boys, and only one has 3 boys. So, the probability is 1/(8-1)

Tougher bonus question: **Dividing time**

If the digits 1 through to 9 are randomly arranged to make a number, what is the probability that that number is divisible by18?

**Answer**: 4/9 or 0.444. Well done to: Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Amparo Botella, Ismael Quesada SA, Spain; Ramasubramanian P, manager, marketing – mixer and LTKMPL products, Larsen & Toubro Ltd, Kanchipuram, Tamil Nadu, India.

Again, Michele Girardi offered a detailed explanation: 18 = 2*3^2, so a number divisible by 18 must be divisible by 2 and 9. Numbers divisible by 2 must terminate by an even number, numbers are divisible by 9 if the sum of their digits is since: 1+2+3+4+5+6+7+8+9 = 45 (= 9×5), all the numbers created by this random arrangement are divisible by 9. Only the ones terminating by 2, 4, 6, 8, are also divisible by 2 and consequently by 18. The probability of this event is 4/9 = 14.3%. Then the total number of permutations is 9!,there are 8! permutations for each of the 4 even final digits, so the probability is 4*8! /9! = 4/9.

Question 1: **Factorial five, plus one**

Complete the following set of factorials:

5! = 5 x 4 x 3 x 2 x 1=120, 4! = 4 × 3 × 2 × 1 = 24; 3! = ?; 2! = ?; 1! = ?; 0! = ?.

**Answer**:

5! = 5 x 4 x 3 x 2 x 1=120; 4! = 4 × 3 × 2 × 1 = 24; 3! = 3 x 2 x 1 = 6; 2! = 2 x 1 = 2; 1! = 1 x 1 = 1; 0! = 1

Well done to all the following readers who avoided tripping over the final step: realising that 0! is not zero:** ****Andrew Beasley**, product analyst, Hankook Tyre UK Ltd, UK; **David Mann**, Polymer Business Development, France; **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **France Veillette**, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada: **John Bowen**, consultant, Bromsgrove, UK; **Hans-Bernd Lüchtefeld**, market research & communication manager, PHP Fibers GmbH, Obernburg, Germany; **Ramasubramanian P**, L&T Rubber Processing Machinery, India; **Thierry Montcalm**, R&D and innovation manager, Soucy Techno, Canada; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria.

Among the many neat explanations provided were Andrew Beasley’s concise definition of n! as n! = n*(n-1)! for n > 0 and this longer one from Ramasubramanian P:

*Mathematically, factorial of a positive integer n is the product of all the integers less than or equal to n and greater than or equal to 1. The significance of ‘factorial’ is that it represents the number of ways ‘n’ elements of a set can be distinctly arranged in a sequence. For example, for a set with three elements, say A,B,C the number of ways this set can be distinctly arranged is 6: ABC, ACB, BCA, BAC, CAB, CBA. Similarly, two elements can be arranged in two distinct ways. A single element can be arranged in only one distinct way. And a set with no elements can be arranged in one distinct way, which is again a NULL set. Hence 0! = 1 too.*

**FEBRUARY**

Great to have a first-time winner of our top award and so well deserved for managing an extra tricky series of teasers this month, including Question 1, which seemed to throw so many of our top players. Huge congratulations go to **Ramasubramanian P** of Larsen & Toubro in India, our new **Brainiac of the Month**.

Question 4: **Country file**

Starting in Turkey and moving west across Europe, find the missing city in the following sequence:

Ankara, _, Oslo, Andorra la Vella.

**Answer**: These are the names (in English) of capital cities beginning and ending with the same letter. The missing city, so, was Warsaw.

Expertly well done to:** John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **David Mann**, Polymer Business Development, France; **Ramasubramanian P**, L&T Rubber Processing Machinery, Larsen & Toubro Ltd, Tamil Nadu, India. A special mention also to **Michele Girardi** Scame Mastaf Spa, Suisio, Italy for his help in pointing out a slight blip with the question on Monday.

Bonus question: **More soup**

As we had two valid answers for Question 3, we ran this on into this week, asking readers to identify what came next in the ‘officially’ correct sequence ABC, BCE, CEH, EHM, _

**Answer**: HMU came next in the sequence, as David Mann explained: gaps of 5 and 8 according to the Fibonnaci series.

Well done to: **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **David Mann**, Polymer Business Development, France; **Ramasubramanian P**, L&T Rubber Processing Machinery, Larsen & Toubro Ltd, Tamil Nadu, India; **Amparo Botella**, Ismael Quesada SA, Spain. A special mention also to **Hans-Bernd Lüchtefeld** of PHP Fibers GmbH in Germany, who came quite close with his answer (pictured across).

Question 3: **Alphabet soup**

Find what comes next in the following sequence:

ABC, BCE, CEH, _

**Answer**: As sometimes happens, a valid, alternative answer has emerged for this week’s Brainteaser. So well done to the following readers who worked out the ‘officially’ correct sequence: ABC, BCE, CEH, EHM: **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **John Bowen**, consultant, Bromsgrove, UK; **Michael Easton**, sales & marketing director, Globus Group, Manchester, UK; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy;

And, equally well done to readers who provided the alternative valid answer EHL: **Amparo Botella**, Ismael Quesada SA, Spain; **Bharat B Sharma**, Sr V.P. product development & technical service (elastomers), Reliance Industries Ltd, Vadodara Manufacturing Division, Petrochemicals, Vadodara, Gujarat, India; **Thierry Montcalm**, R&D and innovation manager, Soucy Techno, Canada; **Ramasubramanian P**, L&T Rubber Processing Machinery, Larsen & Toubro Ltd, Tamil Nadu, India; **Varun Sureka**, Hartex Rubber Pvt. Ltd, Hyderabad, India.

This sequence was explained concisely by Amparo Botella: ABC, BC(D)E, CE(FG)H, EH(IJK)L

Question 2: **Flight time**

A projectile is fired at a horizontal velocity of 150 metres/second. It hits a target at a horizontal distance of 30 metres. Assuming negligible air resistance effect, what is a) the flight time of the projectile? and b) its vertical displacement?

**Answers**: Well done to the following readers who answered part (a) correctly ie 0.2 seconds: **John Bowen**, consultant, Bromsgrove, UK; **Thierry Montcalm**, R&D and innovation manager, Soucy Techno, Canada; **P. Ramasubramanian**, Larsen & Toubro Ltd, Rubber Processing Machinery, Tamil Nadu, India: **David Mann**, Polymer Business Development, France; **Olaf Mayer-Mader**, Abteilung BA-M, product management and business development PA, Festo AG & Co. KG, Denkendorf, Germany.

And even more well done to those readers who answered the more tricky part (b) correctly ie 0.196 metres (0.2m also accepted): John Bowen; Thierry Montcalm and David Mann.

John Bowen’s solution neatly shows how it’s done:

a] Flight time = 30/150 = 0.2 secs

b] Vertical displacement: we use s = ut + 1/2.f.tsqd = 0 [fired horizontally so no initial vertical component] + 1/2 * 9.8*0.2*0.2 [f = acceleration due to gravity, 9.8m/sec/sec] = 0.196 metres.

Question 1: **City sequence**

Fill in the missing cities in the following sequence:

Reykjavík, _, Oslo** …** Montevideo , _, Wellington.

**Answer**: Just two correct replies, so extra well done to: **P** **Ramasubramanian**, manager, marketing – mixer and LTKMPL products, rubber processing machinery, Larsen & Toubro Ltd, Vedal Village, Kanchipuram, Tamil Nadu, India; and **Yuichi (Joe) Sano**, Sumitomo Electric Industries Ltd, Itami, Japan. Our other Brainiacs really do need to get out more.

As P Ramasubramanian nicely explained: This is a sequence of national capital cities sorted by latitudes. Northernmost national capital is Reykjavic (capital of Iceland @ 64.14 N) and the southernmost national capital is Wellington (capital of New Zealand @ -41.28 S).

So, the sequence is:

Reykjavic – capital of Iceland (64.14 N), **Helsinki** – capital of Finland (60.17 N), Oslo – capital of Norway (59.9 N) … Montevideo – capital of Uruguay (34.88 S), **Canberra** – capital of Australia (35.3 S), Wellington – capital of New Zealand (41.28 S).

**JANUARY**

When the going got tough, Brainiacs around the world really got going, and this month’s award could have gone to three or four top contestants. Both for dealing so expertly with all four questions, I am sure everyone will join in congratulating **JOHN BOWEN**, our first **BRAINIAC OF THE MONTH** for 2019

Question 4: **Next number B**

Find the next number in the following series:

123, 354, 897, 1875, 10626, ?

**Answer**: Many thanks to all our readers who had a go at this one – not everyone got to the correct answer, 16887. Extra well done, so, in order of reply to: **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Hans-Bernd Lüchtefeld**, market research & communication manager, PHP Fibers GmbH, Obernburg, Germany; **Bharat B Sharma**, Sr VP product development & technical service (elastomers), Reliance Industries Ltd, Gujarat, India; **Ramasubramanian P**, manager, marketing – mixer and LTKMPL products, rubber processing machinery, Larsen & Toubro Ltd, Vedal Village, Kanchipuram, Tamil Nadu, India; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **John Bowen**, consultant, Bromsgrove, UK: **Andrew Beasley**, product analyst, Hankook Tyre UK Ltd, UK; **Michael Easton**, sales and marketing director, Globus Group, Manchester, UK; **Amparo Botella**, Ismael Quesada SA, Spain; **Yuichi (Joe) Sano**, Sumitomo Electric Industries Ltd, Itami, Japan.

*Lots of neat workings out, but here’s John Droogan’s reply – within minutes of the Monday send-out – to show how it’s done:*

*123 > 231 then add 123 gives 354*

*354 > 543 then add 345 gives 897*

*897 > 978 then add 897 gives 1875*

*1875 >8751 then add 1875 gives 10626*

*10626 > 06261 then add 10626 gives 16887.*

Question 3:** Football focus**

In tests for the next World Cup, a football is modelled as a particle and air resistance is ignored. A player kicks the ball from a height of 0.6m above the ground, propelling it vertically upwards at a speed of 10.5m/s. From the modelling, what is the greatest height above the ground reached by the ball, and calculate the length of time the ball is more than 2m above the ground.

**Answer**: Said it might get tougher and it did with only three correct replies to this week’s teaser, from: **John Bowen**, consultant, Bromsgrove, UK: **Bharat B Sharma**, Sr VP product development & technical service (elastomers), Reliance Industries Ltd, Gujarat, India; **Paul Knutson**, textile engineer, Timken Belts, Springfield, Missouri, USA. Their workings out of the correct answers **6.22 metres** and **1.86 seconds** are each given below:

**John Bowen**

*This problem uses the basic Laws of Motion:*

*i] To calculate height reached – ie when the ball stops in its vertical trajectory we use*

*v sqd – u sqd = 2fs, where f = acceleration due to gravity [ in this case -9.81m/sec/sec] and s = distance, u = initial velocity, v = final velocity, so*

*0 – 110.25 = -2*9.81*s*

*s = 5.62, so total height = 5.62 + 0.6 = 6.22m*

*ii] to determine the time above 2m we need to solve a quadratic of the form s = ut + 1/2f*t sqd where t = time in seconds; s = 1.4 [as it starts at a height of 0.6m]*

*so 1.4 = 10.5*t – 1/2*9.81*t sqd*

*solving for t gives values of 1.99 and 0.143, so the ball is above 2m height for 1.85 seconds*

*Greatest height above the ground reached by the ball = 6.225m*

*Total time the ball is more than 2m above the ground.= 1.857 Sec. (**gravity considered as 9.8m/sq.second)*

**Bharat B Sharma **

*1 Initial height = 0.6 *

*v2 = u2 + 2as v=0 *

*u=10.5m/s *

*(10.5)2= 2×9.8xs a= (-9.8m/sec2) *

*s= (10.5×10.5/2×9.8 )+0.5 m(Initial height) *

*s=5.625m+0.6m **= 6.225m *

*2. Time require to keep ball above 2 m from ground (upward 2m to 6.225m and return)*

*Distance covered 4.225 m upward and 4.225 m down ward *

*Going up time Final speed = 0, s= 4.225 m, a =-9.8m/sq.sec== *

*Initial speed at 2 m = sq root of 2×9.8×4.225= sq rt of 82.81 m/sec*

* =9.1 m/sec *

*Time taken to trave 4.225 m= (V=u+at) =0=9.1-9.8t) *

* = time = 9.1/9.8= 0.9286 sec *

*Going down = distance covered 4.225m, Initial speed is 0 and a=9.8m/sec2*

*v sqare=u sq+2as u=0, a=9.8m/sq.sec, s=4.225) *

*v sq= 82.81 *

*v 9.1m/sec *

*Time for return journey will also be 0.9286sec *

*Total time the ball will be above 2 meter height =2*0.92857= 1.857 sec.*

**Paul Knutson**

*Greatest height = 6.22 m. Time above 2 m is 1.86s. All derived from formula y=v0*t-.5*g*t^2*

Question 2:** Next number A**

** **Find the next number in the following sequence: 10, 9, 17, 50, 199, _

**Answer**: Clues were offered but not really needed as correct replies rolled in, including Michele Girardi’s a(i+1)=a(i)*i-1 making the sequence: 1,10; 2,9; 3,17; 4,50; 5,199; 6, **994**. Well done in order of reply to: **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Andrew Beasley**, product analyst, Hankook Tyre UK Ltd, UK; **John Bowen**, consultant, Bromsgrove, UK: **Dave Stuckey**, Dow, South Wales, UK; **Hans-Bernd Lüchtefeld**, market research & communication manager, PHP Fibers GmbH, Obernburg, Germany; **Bharat B Sharma**, Sr VP product development & technical service (elastomers), Reliance Industries Ltd, Gujarat, India; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Michael Easton**, sales and marketing director, Globus Group, Manchester, UK; **Jon Cutler**, materials development manager, Trelleborg Sealing Solutions, Tewkesbury, UK; **Martin Jones**, customer service manager, network logistics & transport, DHL Supply Chain UK; **Amparo Botella**, Ismael Quesada SA, Spain; **Thierry Montcalm, **R&D manager, Soucy Techno Inc., Canada; **Trey Thies**, growth strategist, engineered performance products, Milliken & Co., Spartanburg, South Carolina, USA; **Yuichi (Joe) Sano**, Sumitomo Electric Industries Ltd, Itami, Japan; **David Mann**, manager rubber technology, SI Group, Béthune, France.

Question 1: **Rubber connect**

Find the rubber connection in: England, Peru, Denmark, Malaysia.

**Answer**: A relatively easy one to start the year, as long you were not thrown off track by ‘Malaysia’ – things might get tougher as we go along. Well done to the following readers who identified EPDM in the country names: **David Mann**, manager rubber technology, SI Group, Béthune, France; **Jose Padron**, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; **John Droogan**, advanced polymers and composites, MegaChem (UK) Ltd, Caldicot, Monmouthshire, UK; **Paul Knutson**, textile engineer, Timken Belts, Springfield, Missouri, USA; **John Bowen**, consultant, Bromsgrove, UK: **France Veillette**, chef environnement, usine de Joliette, Bridgestone Canada Inc. Canada; **Thierry Montcalm**, R&D and innovation manager, Soucy Techno, Canada; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands.

Brainteaser Awards and Results 2018