ERJ Brainteaser: May

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.