# ERJ Brainteaser: December

22 Dec 2023

For consistently stylish and clever answering, its congratulations to **Michele Girardi **of Scame Mastaf, and **Sudi Sudarshan** of Bridgestone Americas – new joint holders of the **Brainiac of the Month** title. Next up the Brainiac of the Year awards...

Question 4: **Elf & Safety**

Santa has 360,000 sacks of presents to deliver in the next 12 hours. Previously, his full-time team of eight elves could prepare that number of sacks, working flat out over this length of time. But new HSE rules require that all workers take a 30-minute break after every two hours on the job. Santa, therefore, has to bring in temporary workers to meet his delivery target.

How many more elves are needed, if the temps work at 25% the speed of the regular crew?

**Answer**: Firstly, apologies for a typo initially in the email address – please kindly resend if your answer did not reach us. Of the replies that did make it through, there was some variation in the answers. The official answer is 7 elves – allowing breaks for the temporary workers – but as it’s Christmas we’ll happily accept other interpretations also given below. Very well done to: **John Coleman**, membership manager, Circol ELT, Dublin, Ireland; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **Sudi Sudarshan**, strategic market & technology insights lead, Bridgestone Americas Inc., USA; **Amparo Botella**, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; **Michele Girardi**, quality manager, Scame Mastaf SpA, Suisio, Italy; **Stephan Paischer**, head of product management and market intelligence, Semperit AG Holding, Vienna, Austria; **Alfred Johnson** (details not supplied, but did do a neat table).

Many thanks for all the nice seasonal messages: wishing readers/Brainiacs also a Merry Christmas and Happy New Year!

SOLUTIONS

**John Coleman**

*Pre-HSE:*

*360k sacks/12hrs = 30k sacks/Hr, or 3,750 sacks/Hr per Regular Elf.*

*Post-HSE:*

*4 x 0.5 Hr breaks over 12hrs, therefore effective time = 12 Hrs - 2 Hrs, or 10 Hrs out of overall 12 Hr period.*

*Therefore, 360k Sacks x 10/12 = 300k sacks prepared by Regular Elves.*

*So, remainder for Temps to prepare = 360k - 300k = 60k over 10 hrs effective, or 6k Sacks/Hr.*

*Temps’ efficiency = 25%, thus 3,750 Sacks/Hr per Regular Elf x 25% = 938 Sacks/Hr per Temp Elf.*

*So, (6k Sacks/Hr) / (938 Sacks/Hr/Temp Elf) means 6.4 Temp elves are required, and we round up to 7 elves to avoid any unnecessary bodily harm - it is Christmas after all!*

**Andrew Knox**

*Previously, 8 regular elves did 12 x 8 = 96 hours of work in 12 hours*

*Now, in the same 12 hours they work 10 hours with 2 hours of breaks. So the regular elves now only do 10 x 8 = 80 hours of work. The 16 missing hours would require 16/10 = 1.6 regular elves or 4 times as many temporary elves (=6.4 temps). So 7 temps in total are required to cover the shortfall.*

**Sudi Sudarshan**

*Presently Santa needs 8 Elves working for 12 hours to complete the sacks of gifts.*

*Total work content = 12*8 = 96 Elf-hours. *

*With the mandatory 30-minute breaks every two hours, they will have four 30-minute breaks or lose a total of 2 hours. Number of Elf-hours worked = 10*8 = 80*

*Required additional work to complete the job on time = 96-80 = 16 Elf-hours at the pace of regular Elves*

*Since the temporary workers work at 25% of regular workers’ speed, they will need 16*4 = 64 Elf-hours to do the job.*

*Assuming that they too are required to take the same breaks as the regular Elves, i.e., work for 10 hours in the day, it will take 64/10 or 6.4 Temporary Elves to complete the task.*

*Rounding up, Santa will need to employ 7 temporary workers to complete putting together the sacks of gifts.*

**Amparo Botella**

*If they have to deliver 360.000 gifts in 12 hours and each elf will work 9 hours instead of 12 (25% less), they will need elves to cover the remaining 24 hours, what means that Santa Clous must bring 4 workers who will work 8 hours each.*

*Ie:*

*1 elf will deliver 33.750 gifts in 9 hours x 8 elves. = 270.000 gifts*

*1 worker will deliver 22.500 gifts (30.000-25%) in 8 hours x 4 = 90.000 gifts*

*Total gifts delivered: 360.000 gifts.*

**Michele Girardi**

*The answer is 8 new elves*

** the nominal rate is 360000/(12x8) = 3750 sack/elf.h*

** with 0,5 h pause every 2h, it's 360000/(15x8) = 3000 sack/elf.h*

** the new elves rate is 0.25x3000=750 sack/elf.h*

*let x be the number of new elves .*

*To produce 360000 sacks in 12 hours :*

*360000 = 3000*12*8+750*12*x*

*x = ( 360000-288000)/9000 = 8.*

**Stephan Paischer**

*The answer is 6,4 temps (rounded to 7).*

*With 360.000 sacks and 8 elves in 12 hours, each elf is working at a speed of 3750 sacks per hour. All elves together finish 30.000 sacks per hour.*

*With 30 min breaks every 2 hours, the effective worktime is reduced to 10 hours.*

*So the gap of to 2 hours translates to 60.000 sacks that need to be covered by temps.*

*The temps work at 25% of the speed, so 937,5 sacks per hour.*

*The 60.000 sacks therefore need 64 temp net working hours to be taken care of.*

*The 64 temp hours are divided by a net working day of 10 hours (also the temps take breaks), so 6,4 temps need to be hired – rounded up to 7.*

**Alfred Johnson **

*Santa will need an additional 7 elves to ensure all presents are delivered on time.*

*At the end, there will be 9 minutes of free time for the 15 elves. I suggest Santa celebrate with a cake-cutting ceremony!*

*Merry Christmas* ?

Our next teaser will be right after the holiday break, over 2-5 Jan, when we will also announce the winners of the highly prestigious **Brainiac of the Year Awards**.

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.*