ERJ Brainteaser: September 2019

Tire designer Simon has a very specific ritual for climbing up the steps to his office. First he climbs up to the middle step and thinks about a new design for 1 minute. Then he climbs up 8 steps and waits until he hears his colleagues chatting. Then he walks down 12 steps and quickly takes one step up. After a short rest, he walks up the remaining steps three at a time which only takes him 9 paces. How many steps are there?

Answer: 49, although those who answered 48 could be correct, as pointed out by some readers. Well done to Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John D Burrows, textile consultant, France; John Bowen, consultant, Bromsgrove, UK; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; Fariha Rashid, marketing analyst, Kraton Polymers LLC , Houston, Texas; Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; Ricardo Azcarate; Ramasubramanian P, manager, marketing – mixer and LTKMPL products, rubber processing machinery, Larsen & Toubro Ltd, Vedal Village,  Kanchipuram, Tamil Nadu, India; Hans-Bernd Lüchtefeld, market research & communication manager, PHP Fibers GmbH, Obernburg, Germany; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada;

Andrew Knox:

If M = middle step, then top step is M+8-12+1+(9*3)= M+24. So there are 24 steps up from the middle, and 24 steps down from the middle, total 48 steps.

For the calculation, the middle step doesn’t count. There is no middle step on any odd number of steps, only on even numbers of steps.

A pie-maker cuts a scalene triangle out of a square sheet of pastry. The longest side of the triangle is 29cms more than the shortest side, and the third side is 15cms longer than twice the shortest side. Each of its sides is a whole number of cms, and the area of the triangle in square cms is twice that of its perimeter in cms. If the original sheet was 30 cms square, what are the lengths of the three sides of the triangle?

Answer: Another tasty teaser, with the answer working out at 10,39 and 35. Very well done to Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Ramasubramanian P, manager, marketing – mixer and LTKMPL products, rubber processing machinery, Larsen & Toubro Ltd, Vedal Village,  Kanchipuram, Tamil Nadu, India; John D Burrows, textile consultant, France; Fariha Rashid, marketing analyst, Kraton Polymers LLC, Houston, Texas, USA; Neha Kaushik, SRF TTB-C, India; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada – and everyone else who had a go.

Special mentions for some fantastic workings out sent in by: Ramasubramanian P, Andrew Knox (shown on wallpaper). Michele Girardi (who pointed out that this can be solved using the Erone/Heron formula area ^2 =  p*(p-a)*(p-b)*(p-c), where p = (a+b+c)/2  (semiperimeter)) and Jose Padron for the following:

From data                                  

Shortest side is 10 cms     x                 

longest side is 39 cms       y = x+29                

third side is 35 cms z = 2x+15              

triangle perimeter is 84 cms         P = x+y+z = x+x+29+2x+15 = 4x+44

triangle area = 168 cms     A = 2P                   

From Heron’s formula, when height is not known

half perimeter          s=½*(x+y+z) = 84/2 = 42

triangle area  A’ = [s(s-x)(s-y)(s-z)]½ = 168

Question 2: Piece of cake

Pete has always had the same number of candles as his age on each of his birthday cakes. If he has had exactly one birthday cake for each year of his life and blown out a total of 861 candles, can you quickly work out how old Pete is?

Answer: This week’s teaser was actually a piece of cake for many, generating a big response from readers who correctly worked out that Pete was 41 years old. Well done and cake all round to: John D Burrows, textile consultant, France; John Bowen, consultant, Bromsgrove, UK: Randa Tharwat, import manager, Nacita Automotive, Cairo, Egypt; Stephan Paischer, head of product management special products,  Semperit AG Holding, Vienna, Austria; Yoganand Nannapaneni, Mascot Systems Private Ltd, Mumbai, India; Fariha Rashid, marketing analyst, Kraton Polymers LLC, Houston, Texas, USA; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Amparo Botella, Ismael Quesada SA, Spain; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada: Jose Padron, material development specialist, Waterville TG Inc., Waterville, Québec, Canada; David Mann, Polymer Business Development, France; Thierry Montcalm, R&D and innovation manager, Soucy Techno, Canada; ; Yuichi (Joe) Sano, Sumitomo Electric Industries Ltd, Itami, Japan; Alan Jackson (no details supplied); Mirco Niklas, process technology, Freudenberg Technology Innovation SE & Co. KG, Corporate R&D, Weinheim, Germany; 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.

Among the neat working out was:

Michele Girardi: The general formula  for the sum of numbers   1..n   is n(n+1)/2 the equation n(n+1)/2 = 861, giving the  solution n=41…

Andrew Knox: Sum of a simple arithmetic sequence of n terms is n/2(2a + (n-1)d), where a is the first term (here 1), and d is the common difference (here also 1), and n=Pete’s age. This simplifies to:

Sum(1:n) = n/2(2+(n-1)), or Sum(1:n)*2 = n*(n+1)

So, 861 *2 = 1722 = 41*42, so n +41

Bharat B Sharma: Series is 1+2+3+4+………n (all natural numbers) Given — 1+2+3+……n = n/2 (n+1) = 861 candles

Equation to solve n2 +n= 1722

n2 +n-1722 =0 and (n+42)(n-41)=0

So n= (-42) or 41

Ramasubramanian P: Let Pete’s age be X. For X birthdays he would have blown 1+2+3+…+X candles.

So,  sum(1:X) = 861

ð  X*(X+1)/2 = 861

ð  X*(X+1) = 1722

ð  X = 41

Question 1: You cannot be series!

What comes next in this sequence: 2, 6, 42, 1806, _?

Answer: Great to see many in such fine form this week – sign of a well-earned holiday. The answer as explained by several refreshed readers was:

2*(2+1) = 6; 6*(6+1) = 42; 42*(42+1) = 1806; and 1806*(1806+1) = 3263442

Very well done in order of reply to: Yuichi (Joe) Sano, Sumitomo Electric Industries Ltd, Itami, Japan; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Bharat B Sharma, Sr VP Product Development & Technical Service (Elastomers), Reliance Industries Ltd, Vadodara, Gujarat, India; John Bowen, consultant, Bromsgrove, UK: David Mann, Polymer Business Development, France; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; –Ramasubramanian P, manager, marketing – mixer and LTKMPL products, rubber processing machinery, Larsen & Toubro Ltd, Vedal Village,  Kanchipuram, Tamil Nadu, India; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada: John D Burrows, consultant, France; Amparo Botella, Ismael Quesada SA, Spain; Yoganand Nannapaneni, Mascot Systems Private Ltd, Mumbai, India; Paul Knutson, textile engineer, Timken Belts, Springfield, Missouri, USA; Mario Swaanen, technical operations manager, Gradient Compounds Netherlands BV, Hilversum, The Netherlands; Ashley Fahey, sustainability principal vice president, Goodyear Pride Network, Goodyear Tire & Rubber Co., Akron, Ohio, USA; Yoganand Nannapaneni, Mascot Systems Private Ltd, Mumbai, India;