ERJ Brainteaser: May
10 May 2024
Each month, ERJ sets a weekly brainteaser, with questions of varying degrees of difficulty. Readers supplying the most accurate (and stylish) answers are then considered for the prestigious Brainiac of the Month title.
Question 2: Alan key
After setting the five-digit code on his bicycle lock, Alan has written out the following reminder:
Added together, the five numbers total 30.
Added together, the third number and the fifth number total 14 .
The fourth number equals the second number plus 1.
The first number is one less than twice the second number.
Added together, the second number and the third number total 10.
What is the code?
Answer: Some impressive mathematical acrobatics here (see Solutions below) to unlock the code 7 4 6 5 8. Very well done to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Bowen, 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; Simon Winfield, director, MacLellan Rubber Ltd, Wednesfield, Wolverhampton, UK; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Maurizio Bolognesi, technical manager, Silikoneurope Srl, Loiano (BO), Italy; Ian Brooker, Sr. Buyer, HB Chemical Corp., a Ravago Company, Twinsburg, Ohio, USA.
SOLUTIONS
John Bowen
Values are as follows:
1st = 7, 2nd = 4, 3rd = 6, 4th = 5, 5th = 8
Initial check: sum = 30
This is solved by setting up a series of simultaneous equations from the data we are given, then making substitutions and solving:
Let the 5 numbers be a,b,c,d,e
a+b+c+d+e = 30 -i
c+e = 14 -ii
d=b + 1 -iii
a=2b - 1 -iv
b+c = 10 -v
Substitute:
a+b+d = 16
a+2b+1 = 16
so2b-1++2b+1 = 16, 4b = 16, so b = 4, and thus d=5, a = 7, c=6, e=8.
Check: a+b+c+d+e=30
Kamila Staszewska
First, all Alan's reminders were transcribed into equations so that the code has ABCDE form, where the letters represent consecutive digits in the code:
A+B+C+D+E=30
C+E=14
D=B+1
A+1=2B
B+C=10
We know that all numbers are digits (0-9). We also know A must be an odd digit because only odd digits can solve the equation A+1=2B.
Therefore, A can be 9, 7, 5, 3 or 1. These 5 options were tested in table below and the other clues were used to calculate the remaining numbers.
One combination of digits adds up to 30. The code is 74658.
|
A |
B |
C |
D |
E |
SUM |
A+1=2B ; |
B+C=10; |
D=B+1 |
C+E=14; |
A+B+C+D+E=30 |
||
9, 7, 5, 3, 1 |
B=(A+1)/2 |
C=10-B |
|
E=14-C |
|
|
Solution 1. |
9 |
5 |
5 |
6 |
9 |
34 |
Solution 2. |
7 |
4 |
6 |
5 |
8 |
30 |
Solution 3. |
5 |
3 |
7 |
4 |
7 |
26 |
Solution 4. |
3 |
2 |
8 |
3 |
6 |
22 |
Solution 5. |
1 |
1 |
9 |
2 |
5 |
18 |
If the target code is made up of v w x y and z, then:
V + w + x + y + z =30
X + z = 14
Y = w + 1
2w = v + 1
W + x = 10
And modelling everything to be only expressed by x:
W = 10 – x
2 * (10 – x) = v + 1 -> v = 19 – 2x
Z = 14 – x
Y = 10 – x – 1 -> y = 11 – x
And if added in the main equation:
19 – 2x + 10 – x + x + 11 – x + 14 – x = 30 -> 4x = 24 -> x = 6
Simon Winfield
If we allocate letter to the numbers A,B,C etc
C+E=14
B+1 = D
2B-1 =A
B+C = 10
Then
A+B+D=16
Substituting what we know for A and D
(2B-1)+B+(B+1)=16
4B=16
B=4
Using this in the other formulae
D=4+1=5
C=10-B= 6
E=14-C=8
A=2*4-1=7
Michele Girardi
Calling a,b,c,d,e the numbers , it's
a+b+c+d+e=30
c +e=14
-b +d = 1
-a+2b = 1
b+c =10
in matrix form
A*X=B
X = inverse(A)*B
using excel
A B
1 1 1 1 1 30
0 0 1 0 1 14
0 -1 0 1 0 1
-1 2 0 0 0 1
0 1 1 0 0 10
inv(A) X=inv(A)*B
0,5 -0,5 -0,5 -0,5 0 7
0,25 -0,25 -0,25 0,25 0 4
-0,25 0,25 0,25 -0,25 1 6
0,25 -0,25 0,75 0,25 0 5
0,25 0,75 -0,25 0,25 -1 8
Sudi Sudarshan
Let the code be ABCDE
A+B+C+D+E = 30
C+E = 14
So, A+B+D = 30-14 = 16
D = B+1
A = 2B-1
B+C = 10
A+B+D = (2B-1)+B+(B+1) = 4B = 16
B = 4
A = 2*4-1 = 7
D = 4+1 = 5
B+C = 4+C = 10 or C = 6
A+B+C+D+E = 7+4+6+5+E = 30 or E = 30-22 = 8
So, the number ABCDE = 74658
Andrew Knox
Formulae directly from the text are:
1) A + B + C + D + E = 30
2) C + E = 14
3) B + 1 = D
4) 2B - 1 = A
5) B + C = 10
From these can be deduced:
1) - 2): A + B + D = 16
Substitute for D from 3):
A + 2B = 15
Substituting for A from 4): 2B -1 + 2B =14, or 4B =16, and B=4
The rest is easy: C = 6, A = 7, D = 5, E = 8
Check in original formulae gives sum all five = 30, etc..
So, the combination code is 74658.
New teaser on Monday
Question 1: Sign language
What should replace X in this sequence?
^, &, *, X,
Answer: The answer ( is at your fingertips, as these are consecutive symbol-keys on a standard computer keyboard. That's the case, at least, in English-speaking countries – so apologies to readers with different keyboard layouts. Very well done, anyway, to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Bowen, consultant, Bromsgrove, UK; Simon Winfield, director, MacLellan Rubber Ltd, Wednesfield, Wolverhampton, UK; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; Tom Jenkins, vice president, business development, R.D. Abbott Co., Inc., Garden Grove, CA, USA; Wong SiauWoon, R&D manager, gloves company, Malaysia; and everyone else who had a go.