ERJ Brainteaser - February

Question 4: Missing a step

                                             1

                              3             6

              ?              4             4

5            7              8             9

Answer: Almost everyone tripped over our ‘missing step’, perhaps because the solution was more straightforward than it seemed. Extra well done, so, to the select few who correctly identified 3 as the missing number: Umit Kaya, Kros Otomotiv, Turkey, Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Woods, technical consultant, Hanau, Klein-Auheim, Germany. And, nice try to everyone else who had a go.

SOLUTIONS

Andrew Knox

If the title "Missing a step" is a clue, then if the missing number is 3, adding the columns gives equal "steps" of 5 units in height (i.e. from left to right 5, 10, 15, 20). A bit weak, but the only pattern in the numbers that sprung to mind. 

John Woods

STAR ANSWER

Sum of the 4 Horizontal Rows

1

9

X + 8

29

- Sum of the 4 Vertical Columns

5

X + 7

15

20

Observations ;

- in the figures 1 to 9 , only 2 has not been used:

 - if used it would give the following sums

 of 1 9 10 and 29 horizontally and 5 9 15 and 20

-> a certain trend towards harmony but not quite there,

Secondly let us chose 3 as the missing link :which would yield ,

-> 1 9 11 29 Horizontally and 5 10 15 20  Vertically which offers improved harmony but not total conformity.

In summary , I propose 3 as the answer.

Please note: Due to staff travel the next teaser will be on 9 March.

There are 8 girls and 12 boys in an acting class. The drama teacher randomly selects three children from the class to come on stage. What is the probability that the teacher chooses at least two boys?

Answer: Not our toughest ever teaser but it proved a challenge for quite a number of readers. No such drama (see Solutions below), though, for: John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain ; John Woods, materials technology consultant, Hanau, Germany. Well done to all and everyone else who had a go.

SOLUTIONS

John Bowen

Good morning, Patrick, stats is not my strong point, but I would approach this one thus:

Initial total number of pupils = 20

Probability of choosing 1 boy = 12/20

Probability of choosing 0 boys is 8/20 [ie all girls]

All other combinations are acceptable so probability of this is [1 - 96/400] = 0.76

Bharat B Sharma

Answer – Probability of having 2 boys selected in 3 artists is 0.6561 (65.61%)

Total Combinations for three children = 20C3 = 20x19x8 divided by 3x2x1 = 1140

Total combination to have either 2 boys and one girl or all three boys

2 boys and 1 girl   = 12C2 x 8C1 = 12x11 divided by 2x1. Multiplied by 8 = 66x8 = 528

All 3 boys == 12C3 = 12x11x10 divided by 3x2x1 = 220

Total combination for 2 boys in 2 artists = 528+220 = 748

Total Combinations = 1140

Total Probability = 748/1140 = 0.6561 (or 65.61%)

Kamila Staszewska

There are 4 possible patterns:

BBB ; P1= 12*11*10/(20*19*18) =19.3%

BBG; P2=12*11*8/(20*19*18)=15.4%

BGB; P same as above

GBB; P same as above

Probability of having at least 2 boys = P1+P2*3 = 1320/6840 + 3*(1056/6840)=4488/6840=~65.6%

If the 2 boys are picked first and they wonder what are the chances for a girl to join them, the answer is 8/18=44.4%!

Andrew Knox

Answer: 0.65614

4 possible outcomes with their probabilities are:

B  B  G   12/20 x 11/19 x 8/18 = 0.154386

B  G  B   12/20 x 8/19 x 11/18 = 0.154386

G  B  B   8/20 x 12/19 x 11/18 = 0.154386

B  B  B   12/20 x 11/19 x 10/18 = 0.192982

Add all four = chance of ca. 0.65614 at least 2 boys are chosen.

Amparo Botella

Total possible ways to choose 3 students

20

3    =1140

Favorable outcomes (at least 2 boys)

-2 boys and 1 girl - 528

-3 boys - 220

Total favorable combinations

528 + 220 = 748

Probability

748 from 1140

748/1140 = 0,656

 John Woods

1. Number of Ways to select 2 boys in the first three choices :

- 20 x 19 x 18 \ 3 x 2 x I = 1140

2 .  2 scenarios exist : either choosing 2 boys and 1 girl or secondly choosing 3 boys and no girls.

Number of Ways   = 12 x 11 /  ( 2 x 1 x 8 ) = 528

-  Second Scenario : 3 boys are selected ,

Number of Ways = 12 x 11 x 10 / 3 x 2 x 1 = 220 

Total  Number of Ways for the total possibilities :

= 528 + 220 = 748

3 . The Probability P is therefore the the Sum of the Possibilities divided by the ,

Total Number of Choices of 2 boys being selected  / Total Number of Ways ->

748 / 1140 = 0,66 rounded up.

New teaser on Monday

Romania, Sweden, Czech Republic, Denmark, __?, __?, Poland.

(Countries in random order)

Answer: This week’s teaser is about the European Union countries that have kept their own currencies rather than join the Euro zone (see Solutions below). Well done to: Hans-Bernd Luechtefeld, Consultant, Germany; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; David Mann, polymer business development consultant, UK; John Woods, materials technology consultant, Hanau, Germany; Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; John Coleman, membership manager, Circol ELT, Dublin, Ireland; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain.

SOLUTIONS

Hans-Bernd Luechtefeld

Hungary is missing.

All are EU members but not joining the EURO.

Kamila Staszewska

The question is actually ‘missing EUR’ - EU members that are outside the Eurozone ?

Romania has leu

Sweden, Czech & Denmark has their krona/koruna/krone

Poland has z?oty

And the missing: Hungary with its forint and Bulgaria with lev.

Andrew Knox

Answer: Hungary.

These are the six EU countries which have not (yet) adopted the Euro as their currency.

David Mann
These are the EU countries which haven’t adopted the Euro.

The two missing from the list are Bulgaria (Lev) and Hungary (Forint).

John Woods

There are 27 member states of the European Union and to date only 6 of them have not adopted the Euro as their national currency. These are :

Denmark - Danish Krone

Sweden - Swedish Krone

Czech Republic - Czech Krone

Romania - Lev

Poland - Zloty

And, finally, my proposed solution is HUNGARY who continue to use the Forint.

This came after looking at their International Phone Code , the Vehicle Registration Plate , Date of Adhesion to the EU, up to my Eureka moment.

Bharat B Sharma

Probably, its linked to EU countries who have not yet adopted the Euro as currency.

Based on the list of European Union (EU) member states that have not yet adopted the euro (as of 2024-2026) are:

Romania, Sweden, Czech Republic, Denmark, Hungary, Bulgaria, Poland

So, the answer is Hungary (uses the forint) and Bulgaria (uses the lev).

Amparo Botella

STAR ANSWER: Lateral thinking

Reply 2

The missing country can be Hungary.

I can see the all the given countries belong to the UE and to the Schengen zone, but do not use the Euro as currency, so the missing country can be Hungary who belongs to the UE and Schengen zone, and do not use the Euro.

Reply 2

Let’s try before the clues, I can think that the capital cities of all these countries have an important river, so the answer could be Hungary with Budapest having the Danube:

Sweden → Stockholm (Baltic Sea / waterways)

Czech Republic → Prague (Vltava)

Romania → Bucharest (Dâmbovi?a)

Denmark → Copenhagen (Øresund strait)

Hungary → Budapest (Danube)

Poland → Warsaw (Vistula)

John Bowen

STAR ANSWER: More lateral thinking

Answer 1: I think the missing country is Germany: 2025 Women’s Euro Group C contained these countries as listed plus Germany.

Answer 2; The missing country is Hungary, also as an EU member but not in the Euro currency zone.

New teaser on Monday.

Barry the barman has filled a 14-pint container with beer. However, he has now received two orders, each for exactly 7 pints. Problem is: he can only find a 9-pint and a 5-pint jug. In how many steps will it take Barry to accurately measure out the order, using the three containers?

Answer: Many readers thought it couldn’t be done, but not – as shown below – this week’s leading Brainiacs: Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Fokko Erdmann, technical product owner, HF Group, Hamburg, Germany; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; John Woods, materials technology consultant, Hanau Klein-Auheim, Germany; John Coleman, membership manager, Circol ELT, Dublin, Ireland. Well done to all and everyone else who had a go.

SOLUTIONS

Bharat B Sharma

Total steps include 13 considering that 14 pint container is already filled with beer!!

Containers used are highlighted by bold letters and underlined. Two containers are used indicating one step.

	14 Pint	9 Pint	5 Pint	TOTAL Pint	Used Containers (pints)
Step 0	14	0	0	14
Step 1	5	9	0	14	14 &9
Step 2	5	4	5	14	9&5
Step 3	10	4	0	14	14 &5
Step 4	10	0	4	14	9&5
Step 5	1	9	4	14	14&9
Step 6	1	8	5	14	9&5
Step 7	6	8	0	14	14 &5
Step 8	6	3	5	14	9&5
Step 9	11	3	0	14	14 &5
Step 10	11	0	3	14	9&5
Step 11	2	9	3	14	14 &9
Step 12	2	7	5	14	9&5
Step 13	7	7	0	14	14 &5

Amparo Botella

The result is 13 steps which are the following ones (lets name the jugs with the number of pints they content)

Fill 9 from 14 → (5, 9, 0)

For Form 9 → 5 → (5, 4, 5)

Empty 5 into 14 → (10, 4, 0)

For 9 → 5 → (10, 0, 4)

Fill 9 from 14 → (1, 9, 4)

Serves 9 → 5 (fills 5) → (1, 8, 5)

Empty 5 into 14 → (6, 8, 0)

For 9 → 5 → (6, 3, 5)

Empty 5 into 14 → (11, 3, 0)

Serves 9 → 5 → (11, 0, 3)

Fill 9 from 14 → (2, 9, 3)

Pour 9 → 5 (fills 5) → (2, 7, 5) The 9-pint jug now contains exactly 7 pints.

Pour the 2 and 5 together in the 14 jugs to get the other 7 pints

Fokko Erdmann

STAR ANSWER: Top newcomer

The following should do the trick for all the thirsty folks: 

Run #	order for filling	content (14 pint glass/9 pint glass/5 pint glass)
1	14 to 9 to 5	5/4/5
2	5 to 14 and 9 to 5	10/0/4
3	14 to 9 to 5	1/8/5
4	5 to 14 and 9 to 5	6/3/5
5	5 to 14 and 9 to 5	11/0/3
6	14 to 9 and 9 to 5	2/7/5
7	5 to 14	7/7/0

John Bowen

I think Barry will need 13 steps as follies:

Step 1: Initial state and first transfer

Barry starts with a 14-pint container (A) full, and empty 9-pint (B) and 5-pint (C) jugs. He fills the 9-pint jug from the 14-pint container.

Containers: (A: 5, B: 9, C: 0)

 Step 2: Fill the 5-pint jug from Jug B

Barry pours beer from the 9-pint jug into the 5-pint jug until it is full.

Containers: (A: 5, B: 4, C: 5)

Step 3: Empty the 5-pint jug into Jug A

He pours the 5 pints from jug C back into container A.

Containers: (A: 10, B: 4, C: 0)

Step 4: Transfer remaining beer from jug B to jug C

He pours the 4 pints currently in the 9-pint jug into the 5-pint jug.

Containers: (A: 10, B: 0, C: 4)

Step 5: Refill the 9-pint jug

He fills the 9-pint jug again using the beer from container A.

Containers: (A: 1, B: 9, C: 4)

 Step 6: Top up the 5-pint jug C

Since jug C already has 4 pints, he pours 1 pint from jug B to fill jug C.

Containers: (A: 1, B: 8, C: 5)

 Step 7: Empty the 5-pint jug into the main container

He pours the 5 pints from jug C back into container A.

Containers: (A: 6, B: 8, C: 0)

 Step 8: Transfer from jug B to jug C

He fills the 5-pint jug using beer from the 9-pint jug.

Containers: (A: 6, B: 3, C: 5)

 Step 9: Empty the 5-pint jug into the main container

He pours the 5 pints from jug C back into container A.

Containers: (A: 11, B: 3, C: 0)

 Step 10: Transfer remaining beer from jug B to jug C

He pours the 3 pints from jug B into jug C.

Containers: (A: 11, B: 0, C: 3)

 Step 11: Refill the 9-pint jug B

He fills jug B using beer from container A.

Containers: (A: 2, B: 9, C: 3)

Step 12: Top up the 5-pint jug C

Since jug C already has 3 pints, he pours 2 pints from jug B to fill jug C. This leaves exactly 7 pints in the 9-pint jug.

Containers: (A: 2, B: 7, C: 5)

 Step 13: Final distribution:

He pours the 5 pints from jug C back into container A, which already has 2 pints.

Containers: (A: 7, B: 7, C: 0)

My local is much more efficient and just ensures an adequate supply of pint pots - much easier! [Yew Tree Inn, Colwall].

Kamila Staszewska

It will take 13 steps.

(Seel also: Sequence of pours presented in neat image above) 

Steps	14pint	9pint	5pint
0	14	0	0
1	5	9	0
2	5	4	5
3	10	4	0
4	10	0	4
5	1	9	4
6	1	8	5
7	6	8	0
8	6	3	5
9	11	3	0
10	11	0	3
11	2	9	3
12	2	7	5
13	7	7	0

John Woods

STAR ANSWER: 'Elbow grease'

The three containers have maximum capacities of 14 liters  , 9 liters and 5 liters respectively or A - B - C  symbolizing 14 - 0 -0 , the initial state . The objective will result in two containers of A and B with 7 liters each and C with zero as it is too low in capacity to contain 7 liters. The objective is therefore 7 - 7- 0.

Initially 14 - 0 - 0

1. A to B : 5 -9 - 0
2. B to C : 5 - 4 - 5
3. C to A : 10 - 4 - 0
4. B to C : 10 - 0 - 4
5. A to B : 1- 9- 4
6. B to C : 1 - 8 - 5
7. C to A : 6- 8 -0
8. B to C : 6 - 3 - 5
9. A to B : 11 - 3 - 0
10.  B to C : 11 - 0 - 3
11.  A to B : 2 - 9 - 3
12. B to C : 2 - 7 - 5
13. C to A : 7 - 7 - 0 -> Objective Attained

To be brutally honest , I have no idea if 13 steps is the optimal answer as I did nothing mathematical and relied on elbow grease and trial and error - thirst quenching work!

John Coleman