ERJ Brainteaser: May

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.

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.