ERJ Brainteaser: July

Question 4: X factor

 1 = 1  
 2 = 2 
 3 = 6
 4 = X

Answer: The Brainiac factor-y made short (and long) work of this one (Solutions below). Very well done to: Hans-Bernd Luechtefeld, consultant, Germany; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK; John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India;  Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Amparo Botella, responsible de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Roshan Samaranayaka, manager - research and development, Jafferjee Brothers Exports (Pvt) Ltd, Colombo, Sri Lanka; P J Swamy, managing  director, Fores Elastomech India Pvt Ltd, Waluj , Aurangabad, Maharastra , India; Dr Kate Burns, senior technologist and regulatory officer, Prisma Colour Ltd, Birch Vale, High Peak, Derbyshire, UK; Sabit Karayel, R&D director, Ozka Lastik, Ba?iskele, Kocaeli, Turkey; and everyone else who had a go.

SOLUTIONS

Hans-Bernd Luechtefeld

The pattern is:

1 = 1 × 1 = 1

2 = 1 × 2 = 2

3 = 1 × 2 × 3 = 6

4 = 1 × 2 × 3 × 4 = 24

Ergo: 4! = 24.

Andy Longdon

Initially, two sequences spring to mind:

Right digit is the product of the previous and current left digits; 1x2=2, 2x3=6, 3x4=12, so X = 12

Or

Right digit is the product of the current left digit and the previous right digit; 2x1=2, 3x2=6, 4x6=24, so X = 24

John Bowen

This ia a Factorial progression:

1 = 1 x 1 = 1

2 = 2 x 1 = 2

3 = 3 x 2 x 1 = 6

4 = 4 x 3 x 2 x 1 = 24

and so on…

Bharat B Sharma

STAR ANSWER: Covers most angles…

I may be wrong but this question appears to be quite simple

X Factor  -- or is it Factorial X??? – which is filling/fitting - 100% in the sequence

• n! = nx(n-1)x(n-2)x…… x(n-n+1)
• 1!= 1               =1
• 2!= 2x1           =2
• 3!= 3x2x1       =6
• 4!= 4x3x2x1   =24
• 5! =5x4x3x2x1=120

• So here the sequence is  1, 2 , 6, 24,120  (answer could be 21)

 Mathematically, there could be a few more models as the sequence has just begun with only three numbers and fourth number can be arrived based on other factors like

1. Liner Pattern---
1. 1
2. 2=  {(1x1)+1}    
3. 6{(2x2)+2}    
4. 21 {(3x6)+3)       
5. 88 {(4X21)+4}

1. So here the sequence could be 1, 2 , 6, 21, 88  (answer could be 21)

1. Perfect Square Difference Pattern

If you evaluate the differences between the numbers as growing perfect squares (\(1^2, 2^2, 3^2...\)): [1]

• 1                                  =1       
• 1+1²     = 2                   =2
• 2+2²     =2+4                =6
• 6+3²     =6+9                =15
• 15+4²   =15+16            =31

1. So here the sequence could be 1, 2 , 6, 15, 31  (answer could be 15)

Andrew Knox

This sequence is the factorial of the numbers to the left of the equals sign, so, X is 4x3x2x1 = 24

Kamila Staszewska,

Factor-IAL is the answer (!)

1!=1

2!=1*2

3!=1*2*3=6

4!=1*2*3*4=24

Amparo Botella

I think the pattern is:

1=1!=1

2=2!=2x1=2

3=3!=3x2x1=6

4=4!=4x3x2x1=24

So X=24

Dr Kate Burns

1 = 1 

 2 = 2 (2x1)

 3 = 6 (3x2x1)

 4 = 24 (4x3x2x1)

I think the sequence this week is the factorials as shown.

 

New teaser on Monday

 

…, _?_ , LXXXIX, CXLIV, CCXXXIII, CCCLXXVII, …

Answer: Venerunt, viderunt, vicerunt (see Solutions below): Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India;  John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Hans-Bernd Luechtefeld, consultant, Germany; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Dr Kate Burns, senior technologist and regulatory officer, Prisma Colour Ltd, Birch Vale, High Peak, Derbyshire, UK; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK; Jose Padron, project coordinator, laboratory R&D, E2Metrix Canada Inc., Sherbrooke, Quebec, Canada; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain;  Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK. Well done to all and everyone else who had a go.

SOLUTIONS

Bharat B Sharma

The sequence appears to be of Roman Numbers—If we decode the Roman number, we get

• LXXXIX = 89
• CXLIV = 144
• CCXXXIII = 233
• CCCLXXVII = 377

Apparent Pattern

The resulting sequence (---, 89, 144, 233, 377)

89+144 = 233

144+233 = 377

Net number , accordingly could be  233+377 = 610

So earlier number (before 89 could be such that its addition will be 144 (144-89==55)

Sequence thus would be == 59, 89, 144, 233, 377,610 etc.

Now as the clues are in Roman numerals, let’s convert 55:

50 = L

• 5 = V
• 55 is LV

The missing letters in the series of numbers in Roman letters  / Answer could be  LV.

John Bowen

Translated into more familiar Arabic numerals, these become 89, 144, 233, 377, a Fibonacci series, ie each number is the sum of the previous two.  So the previous 3 numbers are 21,34,55 or in Roman notation XXI, XXXIV, LV…

Hans-Bernd Luechtefeld

Result is LV (55).

Because:

LXXXIX, CXLIV, CCXXXIII, CCCLXXVII = 89, 144, 233, 377

Adding up 89+144 = 233, 144+233 = 377, etc.

So x+89 = 144, gives x = 55 resp. LV.

Dr Kate Burns

STAR ANSWER: Neat reply

This is the Fibonacci sequence written in Roman numerals. The missing number is LV (55).

Andrew Knox

The preceding number (in Roman numerals) is LV (55 in decimals) and preceding that would be XXXIV (34). See scribbles below.

Reasoning: Taking the differences between successive decimal number equivalents gives the previous number in decimals, which then needs converting back to Roman numerals.

Ancient Romans are still very confused when buying the correct sized underwear – in reference to the modern sizing system of L, XL, etc.. 

Jose Padron

STAR ANSWER: Nice practice run

Hello, this is the Fibonacci series written in Roman Numbers, so the answer is LV = 55

Just to practice my memory, here is the Fibonacci series from the beginning:

I = 1

II = 2

III = 3

V = 5

VIII = 8

XIII = 13

XXI = 21

XXXIV = 34

LV = 55

LXXXIX = 89

CXLIV = 144

CCXXXIII = 233

CCCLXXVII = 377

DX = 510

DCCCVII = 887

MCCCXCVII = 1397

Amparo Botella

The missing term is LV.

Converting the Roman numerals gives:

LXXXIX = 89

CXLIV = 144

CCXXXIII = 233

CCCLXXVII = 377

These numbers are consecutive terms in the Fibonacci sequence:

55, 89, 144, 233, 377, ...

The clue “I, II, III, V…” represents the first Fibonacci numbers written in Roman numerals:

I (1), II (2), III (3), V (5), VIII (8), XIII (13), XXI (21), XXXIV (34), LV (55), LXXXIX (89), ...

Kamila Staszewska 

These are how ancient Romans would written the Fibonacci sequence

…,  LXXXIX, CXLIV, CCXXXIII, CCCLXXVII that’s 89 ,144, 233, 377

55 is before this sequence so LV is the answer

Andy Longdon

Converting these Roman Numerals to modern numbers gives the sequence:

­_?_, 89, 144, 233, 377

Which is a Fibonacci sequence where each number is the sum of the two numbers preceding it. Hence the number missing would be 144-89=55, which in Roman Numerals format is LV.

New teaser on Monday.

_?_, _?_ 125, 2,401, 161,051, 4,826,809, ..

Clues: Prime, Power…

Email your answer: correct replies on Friday.

Bonus question: Find the link

Anaya, Cardini, Grey, Montagu, Wellesley...

Clues: Nachos, Caesar...

Answer: Well done to everyone who grappled with this tough teaser (and bonus Q). Coming out on top were: Bharat B Sharma, technical director TWC Group (Techno Waxchem & Rajsha Chemicals), Kolkata, India; Hans-Bernd Luechtefeld, consultant, Germany; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Dr Kate Burns, senior technologist and regulatory officer, Prisma Colour Ltd, Birch Vale, High Peak, Derbyshire, UK; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Andy Longdon, technical sales manager, Ceetak Ltd, Bedford UK;

SOLUTIONS

Bharat B Sharma

STAR ANSWER: Before any clues were issued

It appears to be a complex series—

There seems to be two Series chains working-

Base series of prime numbers –  2,3,5,7,

This series has exponential powers starting from 1, 2,3,4,5,6 etc.

So the sequence is:

Prime#1 (power 1); Prime#2 (power 2); Prime#3 (power 3); Prime#4 (power 4); Prime#5 (power 5); Prime#6 (power 6);

Which is 2 (power 1); 3 (power 2); 5 (power 3); 7 (power 4); 11 (power 5); 13 (power 6);

21, 32  53, 74 ,  115   136  

which is exactly 2, 9, 125,  2401, 161051,  4826809   

… (with 2 and 9 being the missing two numbers of the series)

Bonus question

Anaya, Cardini, Grey, Montagu, Wellsley...

This was little easier as these were the family names of inventors /imitators of few common food & drinks stuffs

Clue                       Full Name                                                       Inventor / marketer / promoter of

• Anaya:             Ignacio "Nacho" Anaya                                              Nachos.
• Cardini:           Caesar Cardini                                                             Caesar Salad.
• Grey:               Charles Grey (2nd Earl Grey)                                      Earl Grey tea.
• Montagu:        John Montagu (4th Earl of Sandwich)                        Sandwich.
• Wellsley (Wellesley): Arthur Wellesley (1st Duke of Wellington)     Beef Wellington.

Andrew Knox

Answer: the missing numbers to the left are 2, 9.

This sequence is that of consecutive prime numbers raised to the power of ever increasing factors beginning with 1, then 2, etc.. So 125 is 5^3, 2401 is 7^4, and the first two numbers in the sequence are 2^1 = 2, and 3^2 = 9.

Apologies, I had overlooked the bonus question:  Since Wellesley is misspelled [since corrected] in the question, I thought Montague might be as well, and started to look for other Field Marshals... I had no idea that the Earls of Sandwich were called Montegu. The other chaps who apparently gave their names to food-related items were a Mexican chef called Ignaicio Anaya (Nacho's), Earl Grey (a Bergamot-flavoured tea) and Caesar Cardini (who came up with the Caesar salad).

Dr Kate Burns

The sequence this week is based on the prime numbers, where the first prime number is raised to the power 1, the second prime number to the power 2 and so on. As below, the missing numbers are 2, 9.

Sequence Number	Prime Number	Calculation	Answer
1	2	2^1	2
2	3	3^2	9
3	5	5^3	125
4	7	7^4	2401
5	11	11^5	161051
6	13	13^6	4826809

Hans-Bernd Luechtefeld

The answer is 2 (2 exponent 1) and 9 (3 exponent 2).

The series is defined by growing prime numbers, raised by an exponent which increases by 1 each time. The first figure given is 125 (5 exponent 3), so the next below is 3 exponent 2, etc.

Bonus question:

People whose names became associated with famous things resp. food/meals:

Ignacio "Nacho" Anaya → nachos

Caesar Cardini → Caesar salad

Charles Grey, 2nd Earl Grey → Earl Grey tea

John Montagu, 4th Earl of Sandwich → the sandwich

Arthur Wellesley, 1st Duke of Wellington → Beef Wellington

Kamila Staszewska

 The pattern is prime numbers raised to increasing powers:

2^1 , 3^2, 5^3, 7^4, 11^5, 13^6

The answer is 2 and 9

Bonus

These seem to be are surnames of people which foods or dishes were named after(their first names or noble title they held:

Nacho(s) Anaya

Caesar (Salad) Cardini

Earl Gray (Tea)

John Montagu Earl of Sandwich

Arthur Wellesley Duke of Wellington – Beef Wellington

Andy Longdon

The number sequence is prime numbers to the power of their position in the prime number series, so the missing numbers are 1 and 9:

1^1 = 1

3^2 = 9

5^3 = 125

7^4 = 2,401

11^5 = 161,051

13^6 = 4,826,809

For the connection: Anaya, Cardini, Grey, Montagu, Wellesley

These are the surnames of people who have created or inspired famous foods that are named after them:

Ignacio (Nacho) Anaya (Nacho being Spanish nickname for Igancio)

Caesar Cardini (Caesar Salad)

Charles Grey (Earl Grey Tea)

John Montagu (the Earl of Sandwich)

Arthur Wellesley (Beef Wellington).

New teaser on Monday.

109.4, 218.7, 437.4, 874.9, _?_.

Clues: Track, Man in picture, Distance…

Answer: Just a few Brainiacs on the right track this week, and fewer still reached the correct finishing line - 1640.4 yards (Solutions below): Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Coleman, membership manager, Circol ELT, Dublin, Ireland; Dr Kate Burns, senior technologist and regulatory officer, Prisma Colour Ltd, Birch Vale, High Peak, Derbyshire, UK; John Bowen, BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK. But we also accept the alternative interpretations from: 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. Well done to all and everyone else who had a go.

SOLUTIONS

Andrew Knox

These numbers appear to be distances in yards of athletics track events in metres, listed in ascending order and starting with 100 metres. The next distance therefore will be 1640.4 yards (1500 metres), which is indeed the first distance which is not exactly twice the previous distance.

Why the next distance is not 1600 metres is a good question, as this would then have been very close to one mile (which is 1609.3 metres). (Of course, working in furlongs (1/8th mile, or 10 chains) would have been even simpler, one furlong being 201.2 metres, and half a furlong being 100.6 metres).

John Coleman

This is the classic athletics track distances (100 m, 200 m, 400 m, 800 m) converted from metres to yards (1 m ≈ 1.09361 yd).

Therefore, the next distance is the 1,500m (not double!) which equates to our answer - 1,640.4yds.

Dr Kate Burns

This sequence is the length of short and medium-distance running races in yards. The next distance in standard running events is 1500 m, or 1640.4 yards.

109.4 yards = 100 m

218.7 yards = 200 m

437.4 yards = 400 m

874.9 yards = 800 m

1640.4 yards = 1500m

ALTERNATIVELY

BB Sharma

A simple pattern is visible in the main values:

109.4 × 2 = 218.8 → given 218.7 (−0.1)

218.7 × 2 = 437.4 – Given 437.4 (0)

437.4 × 2 = 874.8 → given 874.9 (+0.1)

So, the next number could follow double  with sequence of ((−0.1); 0, +0.1  which can be +0.2)   - (sequence as  -0.1; 0; +0.1; +0.2……)

So the next number would be 2x874.9+0.2 = 1749.8  +0.2  = 1750

Answer is 1750.

Kamila Staszewska

The pattern is race distances 100m, 200m, 400m, 800m expressed in yards.

The last distance is 1600m = 1749.8yd

Hans-Bernd Luechtefeld

1,749.8

These figures correspond exactly to the lengths of the Olympic track events, converted from meters to yards (1 meter ≈ 1.09361 yards): 100 meters = 109.4 yards, 200 meters = 218.7 yards, 400 meters = 437.4 yards, 800 meters = 874.9 yards, 1,600 meters (4 laps around the stadium) = 1,749.8 yards.

New teaser on Monday.