ERJ Brainteaser: October

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...