ERJ Brainteaser - July
29 Jul 2024
Putting all the amazing maths aside, just for once, our top award goes to everyone who answered Q3. Big congratulations to Kamila Staszewska, John Bowen, Michele Girardi, Andrew Knox, Sudi Sudarshan – new joint holders of the Brainiac of the Month title
Question 4: Family squared
A son said to his father: "I will become precisely X years old in the year X squared".
(He gave the value of the whole number X).
The father replied: "Your grandfather made a similar statement to me when I was a boy."
How old was the grandfather when the son was born?
(Grandad's birthday comes before the son's each year)
Answer: The correct answer to this teaser 88 seemed to elude many readers. Very well done, so, to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Bowen, consultant, Bromsgrove, UK; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA, and everyone else who had a go.
SOLUTIONS
Kamila Staszewska
Son will be 45 in 45^2=2025 (he is 44 in 2024).
He was born in x^2-x=2025-45=1980
Assuming grandfather was less than 100 y.o. the year his grandson was born:
1880≤y^2<1980
43.4≤y<44.5 therefore y=44
y^2=1936 - grandfather said to his son he will be 44 in year 1936.
y2-y=1936-44=1892 - year grandfather was born
1980-1892=88 y.o. - age of grandfather when his grandson was born.
John Bowen
Assuming that this takes place in recent years, the son will be 45 in 2025, hence born in 1980
To be within scope, grandfather would be 44 in 1936, hence born in 1892.
So grandfather would be 88 when son was born in 1980.
Sudi Sudarshan
Assuming that this father-son discussion happened recently, X^2 has to be a year where the son would be alive.
45^2 = 2025, 46^2 = 2116, 44^2 = 1936. The only one that makes sense is X=45. So the son was born in 1980
For the father to have heard the same said by the grandfather (except likely in the past tense) he was likely referring to year 1936 when the grandfather would have been 44. So grandfather was born in 1892. So in 1980, grandfather would have been 88 years old.
Amparo Botella
The son will be 45 on 2025 (45x45) so he was born o n 1980
The grandfather was 44 on 1936 (44x44) so he was born on 1892
The grandfather was 88 years old with the son was born.
Andrew Knox
Answer: Grandfather was 88 years old when this (grand)son was born.
Reasoning:
45 squared is 2025, 44 squared is 1936. So if the son would become 45 in 2025, he was born in 1980.
Similarly his grandfather would become 44 in 1936, so he was born in 1936 - 44 = 1892
Grandfather's birthday is before his grandson's so he was 1980 - 1892 = 88 years old.
Michele Girardi
The answer is 88.
Looking for the numbers that have a square between 1800 and 2100, these are 43 to 46 . The year of birth is given by the difference between the square and the number ; as can be seen in the following table , the son is likely born in 1980 and the grandfather in 1892, so the difference is 88
x x^2 birth(x^2-x) age difference
43 1849 1806
44 1936 1892 86
45 2025 1980 88
46 2116 2070 90
New teaser on Monday.
Question 3: Cross country
Complete this list of locations travelled from, by, through and to, on this US trip:
__, Raleigh, Charlotte, Rock Hill, Atlanta, Birmingham, New Orleans, Houston, LA.
Clues: (In italics above)
Question 2: Come fly with me
Materials scientist Gail’s plane from London to Edinburgh takes off at exactly the same time as colleague David’s flight departs Edinburgh for London. A wind current of around 5mph is travelling against the David’s plane. Both planes are travelling at 120 miles per hour relative to the wind (not the ground). If the flight distance is 390 miles each way, how long will it take for the two planes to pass each other mid-flight?
Answer: The planes passed by after around 1hr 38 minutes. Very well done, in order of reply, to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Bowen, consultant, Bromsgrove, UK; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; and also, for coming close, to Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; and Wong SiauWoon, R&D manager, gloves company, Malaysia.
SOLUTIONS
Kamila Staszewska
Gail’s distance: sG= x
David’s distance: sD=390-x
Gail’s speed: VG=120+5=125mph
David’s speed: VD=120-5=115mph
t=s/V
tG= sG/VG=x/125
tD=sD/VD=(390-x)/115
t=tG=tD
x/125=(390-x)/115
0.92x=390-x
1.92x=390
x=203.125mi – They will meet 203.125mi from London.
t=x/125=203.125/125=1.625h=1h37min30s
Andrew Knox
Whatever the headwind might be the two planes are closing in on each other at a speed of 240 mph.
They meet after 390/240 = 1.625 hours. [1hr 37.5 minutes]
John Bowen
London to Edinburgh is 390miles
Let them pass X miles from London, which by definition they must arrive at simultaneously.
David flies at 115mph and Gail at 125mph
Time for Gail to get to X = X/125 hours, Time for David to get here is [390 - X]/115
so [390 - X]/115 = X/125, Rearranging and solving for X, X = 203.125 miles
Gail's time = 203.125/125 = 1.625 hours, David's time is [390 - 203.125]/115 = 186.875/1151.625 = 1.625 hours [check]
So, we can say that they passed after 1.625hours or 1hour 37minutes 30 seconds
Michele Girardi
Hello everybody,
the answer is one hour and 38 minutes.
Considering the wind as a reference system, the planes travel at the same speed and will pass each other
when the sum of the travel distances is equal to the initial distance
2*120*t = 390
so 390/240 h = 1.625 h
Sudi Sudarshan
My answer: 1 hour 39 minutes 34 seconds after departure
Solution: Let S be the distance from London when the two planes meet.
Gail's plane is flying at a uniform speed of 120 mph and David's plane at 115 mph relative to the ground.
Distance covered by David's plane when the two planes cross = 390-S
Let time elapsed be t
Distance = Velocity * Time
S = 120*t
390-S = 115*t
S/120 = (390-S)/115
S = 390*120/235
t = S/120 = (390*120/235)/120 = 390/235 hours = 1 hour 39 minutes 34 seconds
Wong SiauWoon
Flight from London, speed = 120 mph
Flight from Edinburgh, speed = 125 mph ---> 4% faster than Flight from London.
Travel distance = 390 m
From London, meeting time = [(100/204) * 390m] / 120 mph = 1.59 h = 95.4 minutes
From Edinburgh, meeting time = [(104/204) * 390m] / 125 mph = 1.59 h = 96.4 minutes.
Question 1: Two missing
31 |
13 |
18 |
18 |
47 |
16 |
63 |
31 |
54 |
28 |
26 |
26 |
16 |
45 |
61 |
29 |
25 |
? |
86 |
? |
47 |
8 |
39 |
39 |
Answer: Ironically, all but two of our Brainiacs were missing when it came to answering this tricky teaser. So, it’s extremely well done to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; and Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA.
SOLUTION
Kamila Staszewska
The missing numbers are 61 and 36
Number in second column is difference between number in first and third column: 86-25 = 61
Number in fourth column is difference between number in first and second column: 61-25 = 36
The other numbers can be worked out as follows
A |
B |
C |
D |
1 |
b+c |
a-c |
D |
a-b |
2 |
b+d |
a-c |
a+b |
a-b |
3 |
b+c |
a-c |
D |
a-b |
4 |
b+d |
a-c |
a+b |
a-b |
5 |
b-d |
a-c |
a+b |
a-b |
6 |
b+c |
a-c |
D |
a-b |
New teaser on Monday.
Question 1: Larry’s lawn
Landscaper Larry has designed a lawn represented by the triangle ABC, shown in the figure. If length AB is to be 30m long, angle BAC = 70° and angle ABC = 60°, what is the approximate area of the lawn?
Answer: Trigonometry was the key to this tricky gardening teaser, which generated impressive replies (see Solutions below) from: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Wong SiauWoon, R&D manager, gloves company, Malaysia; John Bowen, consultant, Bromsgrove, UK. Very well done to all above and everyone else who had a go.
SOLUTIONS
Andrew Knox
Draw a line perpendicular to the base intersecting at C.
Then height h = 30 / (tan 20 + tan 30) = 31.87 m.
Area = (30 x 31.87) / 2 = 478 m2.
Kamila Staszewska
Area=BC * h/2 ; h=?, BC=?
sin(<ABD)=h/AB ; <ABD=60°
sin(60°)=(√3)/2=h/30
h=√3*15=25.981m
cos(<ABD)=BD/AB
cos(60°)=0.5=BD/30
BD=0.5*30=15m
tan(<CAD)=CD/h ; <CAD=40°
tan(40°)=0.8390996=CD/25.981
CD=0.890996*25.981=21.801m
BC=BD+CD=15+21.801=36.801m
Area = BC*h/2=36.801*25.981/2 = 478.06m^{2}^{}^{}
Sudi Sudarshan
Drop a perpendicular CD from vertex C to base AB.
Let AD = x, then DB = (30-x)
Tan(^A) = CD/AD
Tan(70°) = h/x
2.747 = h/x or h = 2.747x
Tan(^B) = CD/DB
Tan(60°) = h/(30-x)
1.732 = h/(30-x)
h = 1.732(30-x)
2.747x = 1.732(30-x)
x = 1.732*30/4.479 = 11.600
h = 2.747x = 2.747*11.600 = 31.873
Area of triangle = 0.5*b*h
= 0.5*30*31.873 = 478.095
Wong SiauWoon
Angle CAB = 70
angle CBA = 60
angle ACB = 180 - 70 - 60 = 50
30/sin C = BC/sin70
30/sin50 = BC/sin70
BC = (30/sin50)*sin70
BC = (30/0.766) * 0.940
BC = 39.165 * 0.940
BC = 36.82m
Area = 1/2(30*36.82)sin60
Area = 1/2(1104.6)(0.866)
Area = 478.3 m2
John Bowen
The area of any triangle is given by Area = 1/2x base x height
We have a triangle, base 30 metres and height X metres
We need to calculate X, which we do by dividing the lawn into 2 triangles of 70/20/90 degrees, height X and 60/30/90 degrees, height X
The lengths of the two sides opposite the 20 and 30 degrees we shall call a and b, such that [a + b] = 30 [we know this because you told us]
We can now set to with some trigonometry:
Tan 20 = a/X = 0.364, and Tan 30 = [30 - a]/X = 0.577 [from tables] and rearranging:
a = 0.364X and
30 - a = 0.577X
Adding these together: 30 = 0.941X
X = 31.88 metres
so the area is 1/2 x 30 x 31.88 = 478.2 square metres.
Regards, [and credit to my old maths master].
Michele Girardi