# ERJ Brainteaser: March

1 Mar 2021

Some great answering this month, especially for tricky Q3. And, for his mathematical *tour de force* in solving that challenge, it is big congratulations to **Michele Girardi**, Scame Mastaf SpA, new holder of the **Brainiac of the Month** title.

What do these countries have in common?

Czech Republic, Germany, Hungary, Italy, Liechtenstein, Slovakia, Slovenia, Switzerland.

**Answer**: There might well be a mathematical relationship between the order of arrival of the correct answers – border with Austria – and where the replies came from. Special mention also to France Villette, who is based furthest away. Gut gemacht to: **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna; **Daniel Willrich**, redakteur, büro Köln, AutoRäderReifen-Gummibereifung, Köln, Germany; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Federico Fiorini**, managing director, MAI Italia srl, Isola Rizza (VR), Italy; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **France Veillette**: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada.

New teaser on Monday

Sean finds a coloured ticket inside the wrapper of a bar of chocolate. A message says that the card comes in four different colours and encourages him to collect all four. Assuming there is an equal chance of getting any one of the colours, what is the expected number of chocolate bars Sean must buy to get all four?

Answer: This one proved a bit too chewy for most readers, but the answer is 8.33, or nine bars. Extremely well done to: **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Federico Fiorini**, managing director, MAI Italia srl, Isola Rizza (VR), Italy; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy – and everyone else who had a go.

*Solutions*

*Our top trio came up with some really neat answers:*

*Stephan Paischer
For the first colour it’s easy – one box (4/4) as any colour will fit; for the second it’s just 3 out of 4 that will give a suitable color (4/3) as one is already selected from the first round; for the third 2 of 4 (4/2) and for the last one 1 of 4 boxes (4/1).
So 4/4+4/3+4/2+4/1 = 8.33 *

*Federico Fiorini
Sean has collected the first colour from the first bar. Then he’ll have 75% chance to get a different colour from the second one (3 new colours out of 4 colours available) then with a simple proportion he can expect a new colour in 1/0.75=1.33 times.
After that he’ll have 50% chance to get a different colour from the following one (2 new colours out of 4 colours available) then he can expect a new colour in 2 chocolate bars. Once he got the third he’ll have 25% chance to get the remaining colour (1 out of 4) then he will need to eat 4 chocolate bars to get the final one.
Total expected number of boxes = 1 + 1.33 + 2 + 4 = 8.33
That said… there is always that lucky scenario where he will get all 4 colours in 4 boxes only BUT also the very unlikely but not impossible chance he will have to try infinite chocolate bars… *

*Michele Girardi
The exact number is 8.33, as explained below .
The expected number of boxes is the sum of the number multiplied the probability
4*P(4)+5*P(5)+6*P(6) + .....
Let's call the colours A, B, C, D
For N=4 , let's suppose that the last colour to appear is A the probability that A doesn't appear in the first 3 boxes is (3/4)^3, the probability for A to appear in the fourth box is 1/4 But not all the BCD combinations in the first 3 boxes are useful, since one of the colours may be missing, for example C in BBD the total number of cases is 3^3 = 27 the bad ones are :
- the ones with missing B , i.e all the terns based on C and D , 2^3= 8
- the ones with missing C , i.e all the terns based on B and D , 8
- the ones with missing D , i.e all the terns based on B and C , 8
the total is 24, but we must subtract 3, since the combinations BBB,CCC,DDD are counted twice The number of good combinations is 27-21 = 6 the fraction of good combinations is (27-21)/27 or 1-21/27 So the probability P(4, A last) = (3/4)^3* ( 1-21/27)*1/4= 0,023 this can be repeated for B,C and D, so P(4) = 4*0,023 = 0.094 the general formula is P(N)= (3/4)^(N-1) * ( 1 - (2^(N-1)*3-3) / 3^(N-1)
* 1/4 *4
The sum 4*P(4)+5*P(5)+ .....+ N*P(N) converges to 8.3
N P(N) N*P(N) cumulative
4 0,094 0,3750 0,4
5 0,141 0,7031 1,1
6 0,146 0,8789 2,0
7 0,132 0,9229 2,9
8 0,110 0,8818 3,8
9 0,088 0,7960 4,6
10 0,069 0,6924 5,3
11 0,053 0,5873 5,8
12 0,041 0,4893 6,3
13 0,031 0,4023 6,7
14 0,023 0,3275 7,1
15 0,018 0,2645 7,3
16 0,013 0,2124 7,5
17 0,010 0,1696 7,7
18 0,007 0,1349 7,8
19 0,006 0,1069 7,9
20 0,004 0,0845 8,0
21 0,003 0,0665 8,1
22 0,002 0,0523 8,1
23 0,002 0,0410 8,2
24 0,001 0,0321 8,2
25 0,001 0,0251 8,2
26 0,001 0,0196 8,3
27 0,001 0,0152 8,3
28 0,000 0,0119 8,3
29 0,000 0,0092 8,3
30 0,000 0,0071 8,3
31 0,000 0,0055 8,3
32 0,000 0,0043 8,3
33 0,000 0,0033 8,3
34 0,000 0,0026 8,3*

New, maybe easier, teaser on Monday.

Can you find the link between: Theresa May (politics, UK); John McEnroe (tennis, England): Mikhail Gorbachev (politics, Russia); Tigran V Petrosian (chess, World)?

**Answer**: Some great suggested connections but only a select few readers recognised that all four were succeeded by a Boris. Really 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; **Federico Fiorini**, managing director, MAI Italia srl, Isola Rizza (VR), Italy; **France Veillette**: chef environnement, Usine de Joliette, Bridgestone Canada Inc, Canada.

*As Federico Fiorini neatly explained:*

*All four personalities were superseded by a person named Boris in the field + location indicated in brackets.
Theresa May (politics, UK) was followed by Boris Johnson as Prime Minister in UK.
John McEnroe (tennis, England) was followed by Boris Becker as winner of Wimbledon tennis tournament in England.
Mikhail Gorbachev (politics, Russia) was followed by Boris Yeltsin as President of Russia (formerly Soviet Union).
Tigran V Petrosian (chess, World) was followed by Boris Spassky as World Chess Champion.*

What number comes next in this sequence:?

6, 10, 15, 16, 23, _, ...

**Answer**: Our judging panel was highly impressed to see a select number of readers coming up the correct answers, without any clues. The solution was: starting with 1, add the next two prime numbers. Therefore, the next number in the sequence is 24, [6+(7+11)].

Extra well done to: **Michael Easton**, director, Globus Group, Trafford Park, Manchester, UK; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **France Veillette**: chef environnement, Usine de Joliette, Bridgestone Canada Inc, Canada; **Federico Fiorini**, managing director, MAI Italia srl, Isola Rizza (VR), Italy.