# ERJ Brainteaser: October

4 Oct 2020

This month saw a lot of sharp answering by a select few Brainiacs. And using tricky Q1 as the tie-breaker, it’s big congratulations to industry consultant **John Bowen**, **David Mann** of SPC, **Michele Girardi**, Scame Mastaf and **Stephan Paischer**, Semperit – new joint holders of the **Brainiac of the Month** title

There are 120 different ways of arranging the letters, U, K, M, I and C.

If all of these arrangements are listed in dictionary order, starting with in 1st place CIKMU, what numbered place does UKIMC have the list?

**Answer**: As the selection of neat solutions below show, UKIMC comes in at 112th on the list. Very well done, in order of supplying the correct reply. to: **David Mann**, key account manager, SPC Rubber Compounding, UK; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **John Bowen**, rubber industry consultant, Bromsgrove, UK; **Stephan** **Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria;

*Solutions*

*David Mann*

*The group beginning with U is 97 to 120.*

*Then the group beginning UK is 109 to 114, and UKI is 111 to 112, UKIM is therefore 112.*

*Michele Girardi*

*The arrangements that precede UKIMC are :*

*arrangement group quantity
C * * * * 4! = 24
I * * * * 24
K * * * * 24
M * * * * 24
U C * * * 3!= 6
U I * * * 6
U K C * * 2!= 2
U K I C M 1
for a total of 111, so UKIMC is the 112th.*

*For a double check, the arrangements that follow are :
arrangement group quantity
U K M * * 2! = 2
U M * * * 3! = 6*

*indicating the last arrangement as the 120th as known .*

*Andrew Knox*

*Assign numbers to the letters, i.e. C I K M U = 1 2 3 4 5, then it's easy.*

*The first number is then 12345*

*The last number is then 54321*

*There are 6 combinations in the 54... series*

*There are 6 combinations in the 53... series, of which UKIMC (53241) is the third highest, so 8 numbers are higher than 53241. *

*120 - 8 = 112.*

New teaser on Monday

Rubber tech engineer Simon has recently inherited a watch from his late uncle, but finds it to be faulty. The watch gains 6 minutes every hour. Simon set it using his own accurate clock at midnight, and the watch now shows 8:26 am. He knows that the watch stopped 30 minutes ago, so what is the correct time now?

**Answer**: As the stylish solution from Mehmet Koral (below) shows, the answer is 8:10 am. Very well done, in order of supplying the correct reply. to: **John Bowen**, rubber industry consultant, Bromsgrove, UK; **David Mann**, key account manager, SPC Rubber Compounding, UK; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Mehmet Koral**, managing director, C-C Endustriyel LLC, Kadikoy, Istanbul, Turkey.

New teaser on Monday

A football is made by sewing together 12 black pentagonal panels and 20 white hexagonal panels. There is a join wherever two panels meet along an edge.

How many joins are there?

**Answer**: ERJ 21, Readers 7 (including one on the rebound) was the final score, as a surprising number of players shot high & wide in answering this week’s teaser. Not so the following super-strikers who, in order of reply, correctly worked out that the football has 90 joins: **John Bowen**, rubber industry consultant, Bromsgrove, Worcs, UK; **Daniel Willrich**, redakteur, büro Köln, AutoRäderReifen-Gummibereifung, Köln, Germany; **Andrew Knox**, Rubbond International, Ohé en Laak, The Netherlands; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria; **Jose Padron**, material development specialist, Waterville TG Inc. Waterville, Québec, Canada; **David Mann**, key account manager, SPC Rubber Compounding, UK. Well done to all and everyone else who had a go.

*Solutions:*

*John Bowen
Total number of edges is [12 * 5] + {20 * 6] = = 60 + 120 = 180
Each join joins 2 edges, so number of joins is 180/2 = 90.*

*Michele Girardi
The number of edges and vertexes of the separated panels is 12*5+20*6 =
180 ; putting them together, every edge join with another , so the total of joins is 180/2=90 .
Alternative method
Rearranging Euler's formula : Edges = Faces + Vertexes – 2
Faces = 12+20=32
Vertexes = 180/3=60 , since in the solid 3 vertexes join in one
Edges = 32+60-2 = 90*

*Stephan Paischer
Again, offered two approaches:
A – all 20 white panels have 6 sewings each (total 120). This covers already all sewings to the 12 black panels (as no 2 black panels touch each other), but one needs to reduce the double counted sewings between white panels (3 per white panel – 50% of them to eliminate -> 30). So 120 – 30 = 90.
B – all 12 black panels have 5 sewings each (total 60). You need to add all sewings between 2 white panels (3 per white panel, but 50% needs to be eliminated to avoid double counting). So 60 + 30 = 90.*

New teaser on Monday

How many four-digit squares differ by 1 from a multiple of 10?

**Answer**. There seemed to be some variation in interpretation around this question, but the official answer is **27**, as worked out so neatly (see below) by: **John Bowen**, consultant, Bromsgrove, UK; **Michele Girardi**, Scame Mastaf Spa, Suisio, Italy; **David Mann**, key account manager, SPC Rubber Compounding, UK; **Stephan Paischer**, head of product management special products, Semperit AG Holding, Vienna, Austria. Congratulations to all, and everyone else who had a go.

*Solutions:*

*David Mann*

*In the range from 32^2 (1024) to 99^2 (9801) I found 27 squares either 1 above or below a square.*

*John Bowen*

*The smallest number is 1089, which is the square of 33, then 1369 [37] then 1521 [39], 1681 [41], 1849 [43], 2209 [47], 2401 [49] with the units [iex1,x3,x7,x9] being repeated for each 'ten' up to 99, the largest possible with a square of 9801. So there are 27 numbers [3 = 6*4] which differ by 1 from a multiple of 10.*

*Michele Girardi*

*The numbers whose square is a 4-digit number range from 32 to 99 (7 decades)
- the squares that differ by 1 from a multiple of 10 must end by 1 or 9
- this means that the numbers to be squared must end by 1,3,7, or 9
- the squares are, therefore, 7x4-1= 27, where the -1 is to exclude 31.*

New (maybe easier) teaser on Monday...