# ERJ Brainteaser: May

26 May 2023

Some great answering during May, with Q4 as the tie-breaker. Well done, so, to **Michele Girardi, **Scame Mastaf and **Andrew Knox**, Rubbond Int., new joint holders of the **Brainiac of the Month** title.

Question 4: **A number of numbers**

Can you find three numbers such that: The product of the three numbers is a prime; and the second number minus the first number is equal to the third number minus the second number?

**Answer**: There was quite a number of well-considered replies to this tricky teaser, some suggesting that no solution was possible. Undeterred, **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; and **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands cleverly worked out the answer as -3, -1, 1. Well done to both and everyone else who had a go.

Solutions

*Andrew Knox*

*Answer: A = 1, B = -1, C = -3*

*Obviously will be small numbers so need a bit of trial and error to deduce that:*

*If the three numbers in sequence are A, B and C, then if A = 1, B = -1 and C = -3:*

*1 x -1 x -3 = 3, which is a prime number.*

*B - A = -2 = C - B *

*whereby the two set criteria are met.*

* *

*Michele Girardi Solution 1*

*Hello everybody,*

*The solution is -3,-1,1 . *

*the problem is apparently impossible since a prime number can't be the product of 3 numbers , but this can be solved with a trio -x, -1, 1 , where x is a prime number . *

*putting the other conditions *

*-1-(-x) = 1-(-1) *

*x=3 *

* *

*Michele Girardi Solution 2*

*The solution can be any trio -p,-1/x, x where p is a prime number, x is positive and is calculated to satisfy the conditions : *

*-1/x-(-p) = x - (-1/x)*

*-1/x + p = x + 1/x*

*-1 + xp = x^2 +1*

*x^2 - xp + 2 = 0*

*D = p^2-8*

*x = (p± sqr(p^2-8) )/2*

*x>0 is always true*

*here follow the first numbers*

*solution - solution +*

*p -p -1/x x -p -1/x x*

*3 -3 -1 1 -3 -0,5 2*

*5 -5 -2,281 0,438 -5 -0,219 4,562*

*7 -7 -3,351 0,298 -7 -0,149 6,702*

*11 -11 -5,408 0,185 -11 -0,092 10,815*

*13 -13 -6,422 0,156 -13 -0,078 12,844*

Next teaser on Tuesday, due to the UK’s third bank holiday weekend this month.

Question 3: **Grand total**

Can you add eight 8s together to get 1,000?

Answer: As per the clever solutions below, this worked out as: 8 + 8 + 8 + 88 + 888 = 1000. Expertly well done to: **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **David Mann**, key account manager, SPC Rubber Compounding, UK; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Frank Bloemendaal**, manager R&D, Polycomp, Vorden, The Netherlands; **Vasiliy** **Kozlov**, sales engineer, rubber chemicals supplier LLC Benefit-Him, Russia: and everyone else who had a go.

Solutions:

*David Mann*

*There had to be 5 terms for it to end in 0, and one had to be 888 to get near.*

*Michele Girardi *

*To get a sum of numbers terminating by 8 equal to a number terminating by 0, there must be 5 addends or multiple (8*5=40)*

*- that occupates 5 8's *

*- to get a value around 1000, one addend must be 888 , which occupies other 2 8's *

* - the remaining 8 can only be used to form a 88 *

*putting everything together *

*888+8+8+8+88 = 1000*

*or, with algebra : *

*a*8+b*88+c*888 = 1000*

*a+2b+3c = 8*

* trying c=1 *

*a*8+b*88 = 112*

*a+2b = 5*

*trying b=1 *

*a=5-2=3 a= (112-88)/8= 3 *

* *

*Andrew Knox*

*Answer: (In reverse is easiest)*

*1000 - 888 = 112*

*112 - 88 = 24*

*24 - 8 = 16*

*16 - 8 = 8*

*8 - 8 = 0*

*So, 8 + 8 + 8 + 88 + 888 = 1000.*

Question 2: **Runs in the family**

… 7, 5, 8, 6, ?, 3

**Answer**: Firstly, apologies for a problem with the email-reply link and many thanks to everyone who still managed to reply. This week’s tricky teaser referred to the regnal numbers of British monarchs crowned since the last century: So the answer is ‘**2**’, as in: Edward 7, George 5, Edward 8, George 6, Elizabeth 2 and Charles 3. Extra well done to: **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **David Mann**, key account manager, SPC Rubber Compounding, UK; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; and everyone else who had a go.

Question 1: **Who's next?**

VGE, FM, JC, NS, FH, __

**Answer**: These are the initials of the most recent presidents of France: VGE - Valery Giscard d'Estang, 1974-1981; FM - Francois Mitterand 1981-1995; JC - Jacques Chirac 1995-2007; NS - Nicolas Sarcozy, 2007 – 2012; FH - Francois Holland, 2012 – 2017; EM - Emmanuel Macron, 2017-?. Well done to the following readers, especially those who replied in French: **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **David Mann**, key account manager, SPC Rubber Compounding, UK; **France Veillette**: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; **Hans-Bernd Lüchtefeld**, marketing manager, Indorama Ventures Mobility Obernburg GmbH, Germany.