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Question 3: Missing letter

D < G > U  

H < B > N  

E < F > X

C < H > ?

G < C > R

Answer: Strictly speaking the answer is ‘P’ but, as it’s nearly Christmas, we’ll take ‘16’ as well (See Solutions below). Very well done to: Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA; Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; David Mann, key account manager, SPC Rubber Compounding, UK; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Bowen, rubber & tire industry consultant, Bromsgrove, UK; John Coleman, membership manager, Circol ELT, Dublin, Ireland; Peter D. Talbot, research scientist, Chem-Trend LP, Howell, Michigan, USA; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Kenny McCleery, president, WYKO Tire Technology, Davian Enterprises LLC, USA. 

SOLUTIONS

Michele Girardi

Considering the signs between letters as brackets instead of "less than" or  "more than" symbols  and coding each letter by its position in the alphabet, one gets the table

a b c

4 7 21

8 2 14

5 6 24

3 8 ?

7 3 18

it's c/b = a-1   or  c=(a-1)*b :  

21=(4-3)*7

14=(8-1)*2

24=(5-1)*6

18=(7-1)*3

so in the 4th line it's (3-1)*8 = 16, corresponding to P

Sudi Sudarshan

Replacing each letter by its ordinal rank in the alphabet

D < G > U             4 < 7 > 21

H < B > N              8 < 2 > 14

E < F > X               5 < 6 > 24

C < H > ?               3 < 8 > ?

G < C > R              7 < 3 > 18

If x < y < z  is the general form, from the above  z = (x-1)*y e.g. (4-1) * 7 = 21; (8-1)*2  = 14 etc.

So The ordinal number of the missing letter is (3-1)*8 = 16. The answer is P, the 16th letter of the English alphabet.

Stephan Paischer

The answer is P.

All letters from the alphabet are given a number from 1 to 26.

The number of the first letter is reduced by one, then multiplied with the number of the second letter, and the result gives the number of the third letter.

C H P   3 8 16

(3-1) x 8 = 16

16 stands for P.

David Mann

If we transpose the letters to numbers, A=1 etc, then call the first 2 terms A and B, then the third term is (A-1) x B. The missing number is therefore 16.

Andrew Knox

One needs to assign each letter the number according to its position in the current english language alphabet (i.e. A=1, Z=26), then subtract 1 from the first number only in each row, and then substituting "multiply" for < and "=" for >. to get:

3 x 7 = 21

7 x 2 = 14, etc,

and so (for line 4):

2 x 8 = 16

John Bowen

Missing letter is P

By ascribing numbers to letters in alphabetical order [A=1, B =2 etc] the third letter is the product of [first letter -1] and the second letter.

so [5-1] x 6 =24

2 x 8 =16 [P]

6 x 3 =18

John Coleman

Following on from the first clue, it is apparent that the formula required to solve this puzzle is (n-1)*m = X (where format is n < m > x).

Therefore, ? = 16 as (C-1)*H = (3-1)*8 = 16.

Peter Talbot

This was simpler than it first appeared, somewhat embarrassed I needed a clue!

The numbers represent the position order of a letter in the alphabet, A=1, B=2, C-3, etc.

First, replace each letter by its alphabetical number.

These numbers are related by the following equation: (1st integer – 1) x 2nd integer = 3rd integer

In the first example, D (1st)  = 4; G (2nd) = 7; U (3rd) = 21: (4-1) x 7 = 21

For the second example, H (1st) = 8; B (2nd) = 2; N (3rd) = 14: (8 – 1) x 2 = 14

For the third example, E (1st) = 5; F(2nd) = 6; X (3rd) = 24: (5 -1) x 6 = 24

Last clue, G (1st) = 7; C (2nd) = 3; R (3rd) = 18; (7 – 1) x 3 = 18

To determine ?, C (1st) = 3; H (2nd) = 8; therefore ? = (3 – 1) x 8 = 16;

? = P, the 16th letter of the alphabet.

Amparo Botella

The missing letter is P whose position in the alphabet is 16.

C=3 and H=8 ---? 3x8=24-8=16    P=16

Kenny McCleery

For C<H>?...

C = 3

H = 8

(3-1) x 8 = 16

The 16th letter of the alphabet is P.

The missing letter is 'P'.

Question 2: Music & film

What might fill the gap? 

I'm a believer / Life on Mars? (David Jones)

Star Wars / The Deer Hunter (John Williams)

____ / 12 Years a slave (Steve McQueen)

Answer: Okay, not our regular type of question, but we did have a couple of correct answers, showing that it could be solved (especially with those clues added later in brackets). Extremely well done to: Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; and Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA. Thanks to everyone else who had a good go.

SOLUTION

The key is well known people with the same name: so, any movie, or piece of music, that features someone called Steve McQueen would complete the connection.

Stephan Paischer

The clue is that there are always two different people with the same name: David Bowie (born as David Jones) and Davy Jones [of the Monkees]. John Williams (Star Wars theme composer) and John Williams (classical guitarist, whose music featured in The Deer Hunter). Steve McQueen (actor) and Steve McQueen (director of 12 Years a slave). So, The Towering Inferno – as Steve McQueen plays the main character, or: The Thomas Crown Affair, where he also plays the main character.

Sudi Sudarshan

Based on the first clue I figured that the two songs were associated with two different people with the same real name (David Bowie’s real name being David Jones). The second clue confirmed that – the theme music for the two films were played by two different musicians by the name John Williams. However this didn’t tell me why those specific films were chosen. The third clue pointed the common name at Steve McQueen. Steve McQueen is the director of the multiple-Oscar-winning “12 Years a Slave” and shares the name with the famous Hollywood actor. So I looked for films of actor Steve McQueen. My choice is “The Sand Pebbles”, the only film for which actor Steve McQueen was nominated for an Oscar. My favorite Steve McQueen film is The Great Escape though!

Question 1: A good number

Which number when added separately to 100 and 164 can make each a perfect square?

Answer: A good number of replies for this ‘perfect squares’ question, which was a bit more straightforward than recent teasers. That, however, did not stop some Brainiacs bringing higher maths to the party (see Solutions below). Well done to everyone who answered 125, even more so for those who added -64: John Coleman, membership manager, Circol ELT, Dublin, Ireland; John Bowen, rubber & tire industry consultant, Bromsgrove, UK; Mehmet Koral, Erhardt-Leimer representative for Turkey, managing director, C&C Endüstriyel Danismanlik, Egitim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey; David Mann, key account manager, SPC Rubber Compounding, UK; Peter D. Talbot, research scientist, Chem-Trend LP, Howell, Michigan, USA; Michele Girardi, quality manager, Scame Mastaf SpA, Suisio, Italy; Stephan Paischer, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; Sudi Sudarshan, strategic market & technology insights lead, Bridgestone Americas Inc., USA; Jose Padron, laboratory analyst, Toyoda Gosei WTG, Quebec, Canada.

SOLUTONS

John Coleman

125 is the good number as when it is added to 100 & 164 it results in 225 (15^2) and 289 (17^2) respectively.

John Bowen

The number we are looking for is 125, which when added to 100 gives 225 and when added to 164 gives 289, which are both perfect squares, ie squares of squares.

Andrew Knox

Answer: 125

Adding to 100 = 225 = 15 squared

Adding to 164 = 289 = 17 squared

Mehmet Koral

A: Given numbers : 100, 164.

Now, this needs a trial and error solution actually.

Given 100 and that 100 is square of 10. Then I try to add numbers to 100 so that the answer can be a perfect square.

100+21=121 (square of 11) ;  164+21=185 [Not a perfect square (NPS) ]

100+44=144(square of 12) ;164+44=208 [ (NPS) ]

100+69=169(square of 13) ;164+69=233 [ (NPS) ]

100+96=196(square of 14) ;164+96=260 [ (NPS) ]

100+125=225(SQUARE OF 15) ;164+125+289  [ (SQUARE OF 17) ]

Hence the number to be added is 125

David Mann

After a bit of a play with a calculator and excel, I found 2 answers: -64 and +125, which give square numbers of 36 and 100, and 225 and 289.

Peter D Talbot (solo)

125 is the answer.

100 + 125 = 225 = 152; 169 + 125 = 289 = 172.

I must applaud my arithmetic schooling in primary school where at age 8 or 9 I had to learn my “squared” table for all numbers 1-20.

It came in very handy here!

Michele Girardi

The number is 125: there are a few ways to do that .

Calling the number n and  the two squares  a^2 and (a+x)^2 , it's

100+n = a^2

164+n=(a+x)^2

subtracting

(a+x)^2 - a^2 = 64   where a and x are positive integers with a > 10

it's

2ax+x^2=64

a=(64-x^2)/2x

trying , x=1 gives  63/2, which is more than 10 but not an integer , x=2  gives 15 ; substituting  

15^2-100 = 225-100=125

(15+2)^2-164 = 289-164 = 125 

x=3 gives 9,2  which is less than 10, and  x>3 will give still lower values , so x=2 is the only solution

Alternatively without algebra, one may take the series of the perfect squares and  the series of the differences with lag 1, lag 2, .. until you find 64 :

number                       121,144,169,196,225,256,289, 324, ... 

difference lag 1                   23,  25,  27, ...  odd numbers

difference lag 2                           48, 52, 56, 60, 64 OK

difference lag 3                              75, 81, 87, ..> 64

the first  couple is (225,289) , checking :

225-100 = 125

289-164 = 125

Stephan Paischer

The answer is 125.

100 + 125 = 225 = 15²

164 + 125 = 289 = 17²

Jose Padron

…in that way 100+125=225 = 25*25 a perfect square

And 164+125 = 289 = 17*17  a perfect square.

Sudi Sudarshan

My answers: 125 and -64 (question does not say the number has to be positive).

100 + x = a2

164 + x = b2

Subtracting the first equation from the second:

b2 – a2 = 64

(b+a)(b-a) = 64 = 32x2 = 16x4 = 8x8

8x8 = (8-0)(8+0) giving x=0 which is trivial

16x4 = (10+6)(10-6) or b=10, a=6 or x = 36-100 = -64

32x2 = (17+15)(17-15) or b=17, a=15 or x = 152 – 100 = 125

So, both -64 and 125 satisfy the condition.