Question 4: Track & trace
A race car is driven along a straight horizontal test track with constant acceleration. There are three check points A, B and C, in that order, on the road, where AB = 22 m and BC = 104 m. The car takes 2 seconds to travel from A to B and 4 seconds to travel from B to C. What is the acceleration of the car, and the speed of the car at the instant it passes A?
Answer: Acceleration of the car = 5 m/s² and speed at point A = 6 m/s, as neatly worked out (see Solutions from below) by: John Bowen, consultant, Bromsgrove, UK: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Michele Girardi, Scame Mastaf Spa, Suisio, Italy: Mehmet Koral, Erhardt-Leimer representative for Turkey, managing director, C&C Endüstriyel Danmanlk. Well done to all four and everyone else who had a go.
This requires the laws of motion and a little algebra:
Consider A to B: and using s = ut + 1/2 ftsqd, then 22 = u.2 +1/2f.2.2 Eqn 1
and A to C: 126=u.6 + 1/2f.6.6 Eqn 2
Take 3 x Eqn 1: 66 = u.6 + 1/2 f,2.6 Eqn 3
Subtract 2 from 3: 60 = 12f
So F = acceleration = 5 m/s/s
To find initial speed u we substitute in Eqn 1:
22 = u.2 +1/2.5.4 = 2u + 10
u = 6
Initial speed = 6m/s
Basic formulae: v = a.t, d = v.t
So, d1 = t1.(v1 + v2)/2 and d2 = t2.(v2 + v3)/2
We know t1 = 2, t2 = 4, d1 = 22, d2 = 104, so
1) v1 + v2 = 22
2) v2 + v3 = 52
Subtracting 2) - 1) gives:
3) v3 - v1 = 30
We also know:
(t1 + t2). (v1 +v3)/2 = d1 + d2 or 6.(v1 +v3)/2 = 126 so
4) v1 + v3 = 42
Adding 3) and 4) gives 2.v3 = 72 so v3 = 36 and so v1 = 6 m/s
Acceleration a = (v3-v1)/t = 30/6 = 5 m/s/s
The acceleration is 5 m/s^2 and the speed in A is 6 m/s
The input data are :
point A B C
position (x) 0 22 126
time (t) 0 2 6
The equation of motion is
x = 1/2*a*t^2 + v0*t
The simplest way to get a solution is create in excel a graph of position vs. time and fit a quadratic model with intercept = 0
The output is y = 2.5*x^2 + 6x
Matching the terms of the equation of motions we get v0=6 and a=5.
Question 3: Tour de four
Which four-digit number has digit 1 three less than digit 4, which is one more than digit 2, which is twice digit 3, which is not a prime number and is one more than one-third of digit 4.
Answer: 6849, as worked out in wonderfully different ways (see Solutions below) by: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; David Mann, key account manager, SPC Rubber Compounding, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Jose Padron, laboratory analyst, Toyoda Gosei, Waterville, QC, Canada.
Let the four digits be A, B, C, D.
Then A = [D-3]
D = B + 1
B = 2C
C is not prime, AND = [D/3 + 1]
Considering D, D can only have values of 4,5,6,7,8,or 9
and C is not prime AND = [D/3 +1] ; the only values of D divisible by 3 are 6 and 9 thus C = 4 [non prime], and thus D = 9
so A = 6, and B = 8.
So our 4-digit number is 6849.
By playing around with digit 3, 2 doesn’t work but 4 does, so the number is 6849.
The answer is 6849 .
Let's call a,b,c,d the digits, it's
a = d - 3
d = b + 1
b = 2c
c> 1/3 d
c non prime
and of course
0<= a,b,c,d <=9
using c as parameter :
a = 2c-2
b = 2c
d = 2c+1
3c > d 3c > 2c+1 c> 1
The stricter condition for c comes from d = 2c + 1 , combined with d <= 9 , that implies c<=4.
Since c can't be prime this excludes 2 and 3 , we have already excluded 0 and 1, this leaves only 4 .
Substituting in the equations above we get 6849.
For once a problem which can be deduced purely out of one’s head!...
- The 3rd digit must be no prime number – so only the following apply: 1, 4, 6, 8, 9
- As the 2nd digit is twice the 3rd one, the 3rd one must be below 5 – so only the following apply: 1, 4
- For the 2nd digit, only 2 and 8 apply consequently.
- Let’s select 8 for the second digit:
- As the 4th digit is bigger than the 2nd one, the 4th one must be 9.
- And hence the 1st digit must be 6 (three less than the 4th one).
- The solution is 6849.
- Let’s select 2 for the 2nd digit:
- The 3rd one is 1 (half of 2nd).
- The 3rd one must be more than 1/3 of the 4th one, so the 4th one can only be maximum 2.
- For this, the first equation gives a result below zero, so this option doesn’t give a solution.
- So the only solution is 6849.
The only one possible solution to this is: ABCD = 6849
C is not a prime, thus C could be 4, 6, 8 or 9, in that way C can only be 4; because any other digit will not match the other rules.
B = 2C = 2*4 = 8
D = B+1 = 8+1 = 9
A = D-3 = A = 6
C > D/3 thus, C>9/3, C>3 then, C = 4.
Question 2: All in order?
…, ?, 3111, 75, 32.
Answer: 9171. A crude, perhaps, but effective way of throwing even some of our most experienced Brainiacs off the scent – especially as primes feature quite regularly in this weekly teaser. Extra well done, so, to this select trio, who recognised this series of prime numbers written in reverse: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; David Mann, key account manager, SPC Rubber Compounding, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; and everyone else who had a go.
Question 1: Missing letter
F, H, V, W, Y.
Answer: A big ‘hats off’ to John Bowen for recognising straightaway that the missing letter was K, well before any clues were issued. Very well done, so, in order of reply identifying that this teaser was to do with Scrabble (see below) to: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; David Mann, key account manager, SPC Rubber Compounding, UK; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria.
- 10 Points: Q, Z
- 8 Points: J, X
- 5 Points: K
- 4 Points: F, H, V, W, Y
- 3 Points: B, C, M, P
- 2 Points: D, G
- 1 Points: A, E, I, L, N, O, S, T, R, U
Apologies to readers in any country where Scrabble is not so well known or, perhaps, there is a different version.