ERJ Brainteaser: January 2024

Question 3: Missing metal

CaMg, CaS, CaCa (CaSc),  Ca??

Apologies for typo Sc previously given as Cs

Clues: London…

Answer: It did not help that there was initially a typo - apologies, at least that shows our questions are hand-made - in this obscure-looking teaser. So extra well done for decoding the answer Cr (see Solutions below) to: David Mann, Polymer Business Development (retired), UK; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Coleman, membership manager, Circol ELT, Dublin, Ireland; Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; John Bowen, rubber & tire industry consultant, Bromsgrove, UK; Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA; and everyone else who had a go.

SOLUTIONS

David Mann

First stab: Olympic years expressed in atomic numbers?

CaMg = 2012

CaS = 2016

CaCa = 2020

So 2024 = CaCr

Michele Girardi

Cesium should be substituted by Chromium. Then substituting the letters with the atomic numbers, one gets the sequence 2020, 2016, 2020, 2024, 2028

Andrew Knox

If Sc this then refers to the Olympic Games then the answer is Cr for 2024.

John Coleman

Initially, I was rattling my brain to fit Cesium (Cs, 55)) into the equation, but after seeing the typo in the question this morning things became much clearer…

Ca(20)Mg(12) – 2012 Olympics in London.

Ca(20)S(16) – 2016 Olympics in Rio.

Ca(20)Ca(20) (Ca(20)Sc(21)) – 2020 Olympics in Tokyo (covid-delayed until 2021).

So, the next Olympics are this year in Paris, therefore the atomic number in question is 24 = Cr (Chromium).

John Bowen

Missing metal is Chromium,  Cr

Atomic Numbers of metals:

CaMg = 2012

CaS = 2020

CaCa = 2020  [CaSc = 2021]

CaCr = 2024

These refer to years in which Olympic Games were held [or not in the case of 2020 - replaced by 2021 due to covid]

Next is 2024, CaCr, in Paris.

Stephan Paischer

One answer could be Cr (chrome).

If I take the periodic number of the single elements, f.e. Ca 20, Mg 12, S 16, then Cr gives 24, which gives the following series:

CaMg   2012

CaS      2016

CaCa    2020

CaCr     2024

This could be a series of Summer Olympic games (London, Rio, Tokyo, Paris).

Sudi Sudarshan

Thank you for your correction from (CaCs) to (CaSc) as that threw me off!  2055 didn’t make any sense.

Solution: The members of the set are the years in which the Summer Olympics were held, by translating each element’s atomic symbol to its atomic number and concatenating the two numbers.

So CaMg = 2012, CaS = 2016, CaCa = 2020 (CaSc) = (2021)  - of course the originally scheduled Tokyo Olympics were postponed from 2020 to 2021 due to the COVID pandemic.

The next scheduled Summer Olympics are in 2024 in Paris. CaXX = 2024 The element with atomic number 24 is Chromium (Cr).

Question 2: Rush hour

If Lucy drives from her home at a constant speed of 90km/hr, she will arrive at the office 5 minutes early. If the same journey is done travelling at 75km/hr, she will arrive six minutes late. How far is Lucy’s home from her office?

Answer: As was pointed out Lucy has quite a commute at 82.5km - maybe she works from home on some days of the week. Very well done to: Antonella Pagliarulo, product development manager, performance polymer solutions,Thomas Swan & Co. Ltd, Consett, Co. Durham, UK; John Bowen, rubber & tire industry consultant, Bromsgrove, UK; Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA; Peter D. Talbot, research scientist, Chem-Trend LP, Howell, Michigan, USA; David Mann, Polymer Business Development (retired), UK; and (for coming close) Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain.

SOLUTIONS

Antonella Pagliarulo

The distance home-work is given by speed x time. 

The speed we know in both cases, time is x-5min and x+6min, respectively.

The solution is given by solving the system of equations:

First case: 90km/60min · (x – 5min) = y km

Second case: 75km/60min · (x + 6min) = y km

The result is 82.5 km

John Bowen

Lucy's home is 82.5 km from her office.

Calculation:

We need to set up two simultaneous equations.

Lwt distance = d, and time = t hours

Then at 90 kph, d/90 = t - 1/12

and at 75 kph d/75 = t + 1/10

So converting times from hours to minutes then multiplying by 90 these become d = 90t - 7.5   --- Eqn 1

and 1.2d = 90t + 9 ---Eqn 2

Subtracting 1 from 2 gives us

0,2d = 16.5

So d = 82.5km

CHECK:

At 90km/hr, time = 82.5/90 hr = 55 minutes

and at 75 km/hr, time = 82.5/75 = 66 minutes

as my old maths master would say, 'QED'

Stephan Paischer

If I define the travel time as x for speed of 90 km/h, and to (x+11) for speed of 75 km/h, and then calculate the distance by multiplying speed (in km/h) with travel time (x/60, in hrs):

(90 * x) / 60 = ((75 * (x+11) / 60

X = 55

Travel times are 55 mins for 90 km/h, and 66 mins for 75 km/h.

The distance is 90 * 55 / 60 = 82,5 km, or 75 * 66 / 60 = 82,5 km.

Andrew Knox

90 kph = 1.5 km/min, 75 kph = 1.25 km/min

Distance to office D = 1.5(t-5) = 1.25(t+6)

So, solving for t,  t = 60, and D = 90 x 55/60 = 82.5 km

(Similarly D = 75 x 66/60 = 82.5 km)

Michele Girardi

t = s*(1/v)

t1 = s*(1/v1)     t2 =s*(1/v2)

t2-t1 = s*(1/v2-1/v1)

11/60 =s*(1/75-1/90)

s = 82.5

Sudi Sudarshan

Let x be the distance in km from Lucy’s home to her office and T the time in minutes taken to reach the office on time.

Time taken in minutes at a speed of 90 km/h  = x*60/90 =2x/3 = T-5

Time taken in minutes at a speed of 75 km/h = x*60/75 = 4x/5 = T+6

Subtracting the first equation from the second: 4x/5 – 2x/3 = (T+6)-(T-5)

2x/15 = 11  or x = 15*11/2 = 82.5 km

That’s quite a commute!

Bonus: Time taken to arrive on time = 2*x/3 + 5 = 2*82.5/3 + 5 = 55+5 = 60 Minutes or 1 hour

David Mann

This took more thought than I expected. Schoolboy error, forgetting to change the units from Km / h to Km / minute!

Speed = distance / time

Let the speeds be S1 (90 km / h ) and S2 (75 km /h ), the times T1 and T2, and the distance D.

T1 = (D/S1)*60

T2 = (D/S2)*60

T2 - T1 = 11

so

(D/75)*60 - (D/90)*60 = 11

Multiply by 450

6D- 5D = 450 * 11 / 60

D = 82.5 km

Peter D. Talbot

Let x = distance of Lucy’s home from work

Let t = time taken to travel from home to office in hr. NOTE: 5 min = 0.0833 hr; 10 min = 0.1 hr.

x/90 = (t - 0.833), x = 90(t – 0.833)

x/75 = (t + 0.1), x = 75(t + 0.1)

Therefore, solving for t,

90t – 15 = 75t + 7.5, 

t = 1

Therefore, solving for x,

 x = 90 – 7.5 = 82.5km

Alternatively,

x = 75 + 7.5 = 82.5km

Further checking answers,

82.5km @ 90km/hr = 0.9167 hr = 55 minutes

82.km @ 75km/hr = 1.1 hr = 66 minutes

Amparo Botella

The distance between Lucy home and the office is around 81 km.

Using the formula of distance=speed * time, and converting minutes into hours:

D=90*(T-0,08)

D=75*(T+0,1)

90T-7,2=75T+7,5

15T=14,7

T=0,98

D=90*(0,98-0,08)=81

D=75*(0,98+0,1)=81

Question 1: Tricky triangle

What are the dimensions of the square in this triangle, if the length of the horizontal base is 4cm, the vertical side is 3cm and the hypotenuse is 5cm?

Answer: Quite a tester after the holiday break, so very well done (Solutions below) to: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; David Mann, Polymer Business Development (retired), UK; Amparo Botella, procurement & quality, Ismael Quesada SA, Elche, Alicante, Spain; Alfred Johnson, project specialist, AlMailem Tires, Kuwait; Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA; and everyone else who took off their party hat to have a go.

SOLUTIONS

Andrew Knox

The sides of the square are each approx. 1.7142857 (12/7).

Let the sides of the square be x, then the area of the square is x2, and the areas of the two small triangles remaining are x.(3-x)/2 and x.(4-x)/2.

We know the area of the outer triangle is 3.4/2 = 6

So, x2 +  x.(3-x)/2 +  x.(4-x)/2 = 6, or, simplified, 7/2.x = 6 so x = approx 1.7142857

David Mann

Consider the triangle with base A2                       

Height is O2                     

O2 = A1 (sides of a square)                       

As the big triangle is base 4, height 3, then O2 = A2 x 0.75                        

But O2 = A1 so A1=A2 x 0.75                    

A2 = A1 / 0.75                  

A1 + A2 = 4                        

So A1 + (A1 / 0.75) = 4                  

Multiply by 0.75                              

1.75 A1 = 3                        

A1 = 1.714                         

The square has sides of 1.714.

Amparo Botella

The result of the first brainteaser of the year is: each side of the square is 1,714 cm.

A= 3   A=x+(3-x)

B= 4   B=x-(4-x)

(3-x)*x +  (4-x)*x + x2 = 6   (el area de los dos triangulos pequeños + el area del cuadrado es igual al area del triangulo grande que es 3*4/2=6)

      2               2

3x-x2 + 4x-x2 + x2 = 6

   2             2

3x-x2+4x-x2+2x2=12

7x-2x2+2x2=12

7x=12

X=12/7= 1.714

Alfred Johnson

The sides of the square will be [around 1.8 cm] each. I couldn't solve it with the traditional trigonometry, however, MS powerpoint came for help...!

Sudi Sudarshan

My answer: 12/7 cm x 12/7 cm

Solution:

Let x be the side of the square.

 Based on proportionality of triangles,

 x/(3-x) = 4/3 

3x = 4(3-x) = 12 – 4x

7x = 12

X = 12/7