ERJ Brainteaser: August

Question 4: Magic numbers

10 + 9 = 52

8 + 7 = 36

6 + 5 = 24

4 + 3 = 12

2 + 1 =  ?

Answer: Bit of a double-bluff this week in that our question-setters did again go with a prime-number question. Very well done to Hans-Bernd Luechtefeld, consultant, Germany;  Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Amparo Botella, responsible de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain. Also impressive were the alternative answers from: Kamila Staszewska, R&D / Quality Lead, MSc (Eng), Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; and Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK.

SOLUTIONS

Hans-Bernd Luechtefeld

The pattern treats the numbers as ordinal positions of prime numbers (rather than ordinary addition):

1st prime = 2

2nd = 3

3rd = 5

4th = 7

5th = 11

6th = 13

7th = 17

8th = 19

9th = 23

10th = 29

Thus:

10 + 9 → 10th prime + 9th prime = 29 + 23 = 52

8 + 7 → 19 + 17 = 36

6 + 5 → 13 + 11 = 24

4 + 3 → 7 + 5 = 12

2 + 1 → 3 + 2 = 5

B B Sharma

I think, this month is more to do with prime numbers.

This is also tricky one considering the fact that there is hardly any linkage between the sequence given.

However if we see the prime number list and consider the left habd side of equation as the particular prime number (i.e. 10+9 taking as 10th prime number and 9th prime number and add, we get the answer (52).

Pl see the explanation given below for the full sequence applying above logic- --

Prime number--          1st 2;                2nd 3;               3rd 5;                4th 7;                5th 11;     

                                    6th 13;              7th 17;              8th 19;              9th 23,              10th 29

10 + 9  (10th prime (29) + 9th prime (23)         = 29+23   = 52

8 + 7    (8th prime (19) + 7th prime (17)           = 19+17   = 36

6 + 5    6th prime (13) + 5th prime (11)            = 13+11   =24

4 + 3    4th prime (7) + 3rd prime (5)                = 7 + 5      =12

Applying the same logic :

2 + 1    2nd prime (3) + 1st prime (2)                = 3 + 2      =5

Andrew Knox

These are the sum of the pairs of prime numbers to the left of the equations whereby the numbers to the left are the 10th, 9th, down to the 1st prime number, the first 10 prime numbers being 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. So 29+23=52 and the answer here 2+3=5.

Amparo

The numbers given on the left are positions of prime numbers, not the numbers to be added directly.

10 + 9 → 10th prime + 9th prime

= 23 + 29 = 52

8 + 7 → 8th prime + 7th prime

= 19 + 17 = 36

6 + 5 → 6th prime + 5th prime

= 13 + 11 = 24

4 + 3 → 4th prime + 3rd prime

= 7 + 5 = 12

2 + 1 → 2nd prime + 1st prime

= 3 + 2 = 5

So the answer is 5. 

ALTERNATIVELY

Kamila Staszewska

My interpretation is based on using only even numbers of each equation (10, 8, 6, 4, 2).

First multiplying each number by 4, then add number 4 multiplied by sequence of odd numbers (3, 1, 0, -1, -3).

This gives as follows:

10 + 9 = 52  -- > (10 * 4) + (4 * 3) = 40 + 12 = 52

8 + 7 = 36 -- > (8 * 4) + (4  * 1) = 32 + 4 = 36

6 + 5 = 24 -- > (6 * 4) +(4 * 0) = 24

4 + 3 = 12 -- > (4 * 4) + (4 *-1) = 12

2 + 1 = ? -- > (2 * 4) + (4 * -3) = -4

Andy Longdon

So, if we take the row number and multiply by the first number of the pair, and then add the second number, and then a third number, let’s call it y, that reduces by a fourth number z. To make the first two rows work, initial value of y is 33, and initial value of z is 20; z then reduces initially by 12, with the reduction for subsequent rows halving each time compared to the last:

(1 x 10) + 9 + y1 = 52 (y1 = 33 to start) : 10 + 9 + 33 = 52

(2 x 8) + 7 + y2 = 36 (z1 = 20 to start ; y2 = y1 – z1 = 33 - 20 = 13) : 16 + 7 + 13 = 36

(3 x 6) + 5 + y3 = 24 (z2 = z1 -8 = 12 ; y3 = y2 – z2 = 13-12 = 1) : 18 + 5 + 1 = 24

(4 x 4) + 3 + y4 = 12 (z3 = z2 – ((z1-z2)/2) = 12 – (20-12)/2 = 12 – 4 = 8 ; y4 = y3 – z3 = 1 - 8 = -7) : 16 + 3 -7 = 12

(5 x 2) + 1 + y5 = ? (z4 = z3 –((z2-z3)/2) = 8 – (12-8)/2 = 8 – 2 = 6 ;  y5 = y4 – z4 = -7 – 6 = -13) : 10 + 1 – 13 = -2

? = -2

 New teaser on Tuesday – due to a UK bank holiday.

10, 11, 101, 111, ?

Answer: Seems primes in binary code are not everyone’s cup of tea... but bread & butter to top Brainiacs: Kamila Staszewska, R&D/quality lead, MSc (Eng), Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Hans-Bernd Luechtefeld, consultant, Germany; Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK; Dr Kate Burns, senior technologist and regulatory officer, Prisma Colour Ltd, Birch Vale, High Peak, Derbyshire, UK. Well done to all and everyone else who had a go.

SOLUTIONS

Kamila Staszewska

These are primes in binary: 10 = 2, 11 = 3, 101= 5, 111=7, next number is 11 which is 1011 in binary.

Hans-Bernd Luechtefeld

The numbers are represented in the binary system:

10? = 2 → prime number

11? = 3 → prime number

101? = 5 → prime number

111? = 7 → prime number

So the next one is: 1011? = 11??.

Bharat B Sharma

Answer- next numbers in sequence will be 1011, 1101.

The basic instinctive response goes that this is Binary coding of 0 and 1

The numbers written as

10, 11, 101, 111,  are simply the prime numbers  2,3,5,7 respectively and the next prime numbers would be 11,13, etc.

If we convert these numbers in Binary systems , these can be written as 1011, and 1101 respectively.

So the sequence will be

10, 11, 101, 111, 1011, 1101…

Andrew Knox

Answer: 1011

These are the first five prime numbers (2, 3, 5, 7, 11) in the decimal system represented here as binary numbers.

John Bowen

STAR ANSWER: Possible alternative 

Next is 1001 These are binary equivalents of 2,3,5,7; next is 9 or 1001 in binary notation [8 + o + 0 + 1]. 

Alternatively:

Alternative answer is 1011, which is 11 in hexadecimal, as the series is then of prime numbers. [2,3,5,7,11, 1 is not prime by definition]

Andy Longdon

The most obvious answer is that these are decimal prime numbers expressed in binary form, although I cannot see the relevance to the “Give me five!” clue, other than perhaps these are the first 5 prime numbers?

10 = 2

11 = 3

101 = 5

111 = 7

Next (the 5th) prime number = 11, which is 1011 in binary. So next in the sequence is “1011”

Dr Kate Burns

I think the sequence this week is the prime numbers written in binary.

10 = 2

11 = 3

101 = 5

111 = 7

1011 = 11

New teaser on Monday

318, 342, 366, 474, 586, 826, ?

Answer: This week’s teaser proved quite a challenge for all but a select group of our top Brainiacs. Extra well done to: John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Hans-Bernd Luechtefeld, consultant, Germany; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford, UK; and everyone else who had a go.

SOLUTIONS

John Bowen

STAR ANSWER: Twigged it straightaway

Each number is the sum of the previous one plus the product of their digits:

318 + 3 x 1 x 8 = 318 + 24 = 342 and so on, so next here is 826 + 8 x 2 x 6 = 826 + 96 = 922

Bharat B Sharma

Difference between the two digits = first number plus product multiplication of its digits

24 (342-318) - which is product of digits of 1st number of the two numbers(3x1x8) = 318+(3x1x8) = 342
24 (366-342) -which is product of digits of 1st number of the two numbers (3x4x2) = 342+(3x4x2) = 366
108 (474 -366) - which is product of digits of 1st number of the two numbers (4x7x4) = 366+(3x6x6) = 474
586 (586-474) - which is product of digits of 1st number of the two numbers (5x8x6) = 474+(4x7x4) ==586
240 (826-586) - which is product of digits of 1st number of the two numbers (5x8x6) = 586+(5x8x6)= 826
Logically, the next number should be 1st number (826) +(8x2x6) which will be 826+96 = 922
Thus the next number in series of 318, 342, 474, 586, 826, will be 922.

Hans-Bernd Luechtefeld

Each number increases by the product of its own digits (3 × 1 × 8 = 24, 3 × 4 × 2 = 24, etc.) and adding the final product 8 × 2 × 6 = 96 to 826 gives 922.

Sequence Breakdown

318 + (3 × 1 × 8 = 24) = 342

342 + (3 × 4 × 2 = 24) = 366

366 + (3 × 6 × 6 = 108) = 474

474 + (4 × 7 × 4 = 112) = 586

586 + (5 × 8 × 6 = 240) = 826

826 + (8 × 2 × 6 = 96) = 922

Kamila Staszewska

The sequence is first number plus digits multiplied, ie:

318 + 3*1*8= 318 + 24 =342

And so on. Last number is:

826 + 8*2*6= 826 + 96 = 922

Andy Longdon

The next number in the sequence is the current number, plus the product of all three digits of the current number:

318

342 = 318 + (3 x 1 x 8) = 318 + 24 = 342

366 = 342 + (3 x 4 x 2) = 342 + 24 = 366

474 = 366 + (3 x 6 x 6) = 366 + 108 = 474

586 = 474 + (4 x 7 x 4) = 474 + 112 = 586

826 = 586 + (5 x 8 x 6) = 586 + 240 = 826

? = 826 + (8 x 2 x 6) = 922

Missing number = 922

New teaser on Monday.

Q1 - EXTRAS

With apologies, we omitted to include these splendid replies from Andrew Knox of Rubbond International, The Netherlands to Q1: All out!

...158, 160, 161, 164, 167, ?. (Not a mathematical sequence)

Clues: 3 X 20: 3 X 20; …

(Not a mathematical sequence)

Answer: Apologies to most readers, as this teaser did require some knowledge of scoring in darts. That said, we did have brilliant correct answers (Solutions below) from: Hans-Bernd Luechtefeld, consultant, Germany; followed soon after by Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK. Extra well done to both, and everyone else who had a throw. 

SOLUTIONS

Hans-Bernd Luechtefeld

STAR ANSWER: Early day 1 - Brilliant!

Correct answer: "The Big Fish" ?

Thank you for your clue "All out". In darts, a high finish (or high checkout) is a checkout of 101 or more in a single visit of three darts.
The highest scores with typical route:

170 = T20 – T20 – Bull ("Big Fish")

167: T20 – T19 – Bull

164: T20 – T18 – Bull

161: T20 – T17 – Bull

160: T20 – T20 – D20

158: T20 – T20 – D19

157: T20 – T19 – D20

156: T20 – T20 – D18

155: T20 – T19 – D19

154: T20 – T18 – D20

153: T20 – T19 – D18

152: T20 – T20 – D16

151: T20 – T17 – D20

150: T20 – T18 – D18

Andy Longdon

In darts, these are 3 dart check-out possibilities where the final dart must be a double or the bullseye:

...158, 160, 161, 164, 167, ?.

 158 = treble 20, treble 20, double 19 or treble 20, treble 16, bull

160 = treble 20, treble 20, double 20

161 = treble 20, treble 17, bull or treble 19, treble 18, bull

164 = treble 20, treble 18, bull or treble 19, treble 19, bull

167 = treble 20, treble 19, bull

So 170 is the missing number and highest possible 3 dart check-out:

170 = treble 20, treble 20, bull. 

New (more open) teaser on Monday.