ERJ Brainteaser: June

The government has put a new 15% tax on 20” ultra-high performance (UHP) tires this month. This follows a 4% sales tax in May, a €15 recycling charge in April and a 10% carbon tax in March.

If the UHP tires now cost €160, approximately how much did they cost before the carbon-tax increase in March?

Answer: Tough, it seems, to be a tire dealer these days. But our top experts here can, at least, help with the maths. For working out the correct answer as €108 (€112 also accepted, see Solutions below), it’s very well done to: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Michael Easton, director, Globus Group, Trafford Park, Manchester, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Flemming Vorbeck, Nordic purchasing manager commercial, NDI Group A/S, Brørup, Denmark; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc.; France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; Sergey Krasnozhon, team leader, SIBUR International GmbH, Vienna, Austria; Michael Kazas, sales development manager, Bridgestone/Bandag Tyre Solutions, Greece; Jose Padron, material development specialist, Waterville TG Inc. Waterville, Québec, Canada; and everyone else who had a go.

Solutions

John Bowen
Let X = original price in Euros; then this reduces to a single equation:
[[1.10X +15]1.04]1.15 = 160
Solving for X: X = 107.98 or €108

Flemming Vorbeck

Rohit Kale
There are two possible answers, based on whether there would be tax on tax or not?
If there is simple tax on tax, then the price in Marc before Carbon tax will be €108. And if we have to avoid the tax on tax, then it would be €112.

Stephan Paischer
This is considering that the applicable tax base is the sales price of the relevant month, including taxes.
In case we consider that only the base price is taxable (and not the added taxes neither the recycling fee), and that before the carbon tax there were no taxes applied as such, then the answer is €112 .

Sergey Krasnozhon

Jose Padron 
Hello, the UHP tire cost before March was € 107.99 (€ 108.00 in rounded numbers) Here are the figures:

 

New teaser on Monday

 

Working from left to right across each line, which numbers are represented by X and Y in the final line?

4   8   3   2
6   8   5   4
7   6   4   1
9   7   X   Y

Answer The answer is 7 and 5, as Stephan Paischer neatly explained: The digit sum of the first and second number (for a two digit number another digit sum is done) gives the third number, and the digit sum of the second and third gives the fourth number (see also Solutions below). Extremely well done to table-toppers: Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Jose Padron, material development specialist, Waterville TG Inc. Waterville, Québec, Canada; Wolter A. Veenhoven, Hastex International kk, Takenomaru, Naka-ku, Yokohama, Kanagawa, Japan; Hans-Bernd Lüchtefeld, marketing manager – automotive safety, Indorama Mobility Group, PHP Fibers GmbH, Obernburg, Germany; Sergey Krasnozhon, team leader, SIBUR International GmbH, Vienna, Austria; and everyone else who had a go.

Solutions:

Sergey Krasnozhon 

Hans-Bernd Lüchtefeld

Jose Padron

 

New teaser on Monday.

 

Statistician Stan puts nine counters into a bag. There is an even number on three of the counters, while the other six counters carry an odd number.

Stan takes three counters at random from the bag and adds the numbers on the counters together to get a total. What is the probability that the total is an odd number.

Answer: Statistically speaking, probability questions make for the toughest teasers: generating the most off-the-mark replies even from some of our best Brainiacs. So, extra well done to the following select set, who correctly answered 0.45 or 45%: Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Bharat B Sharma, advisor, Technical Service (Elastomers), Reliance Industries Ltd, Vadodara Manufacturing Division, Vadodara, Gujarat, India; France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; Sergey Krasnozhon, team leader, SIBUR International GmbH, Vienna, Austria; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria.

Solutions

Michele Girardi
When you do the sum of 3 numbers, the result is odd if one is odd or 3 are odd :
P(total is odd )=P(1 odd) +P(3 odd)
the single odd number may occur in the first, the second or the third counter
P(odd/even/even) +P(even/odd/even) + P(even/even/odd)+ P(odd/odd/odd) = = +6/9*3/8*2/7 +3/9*6/8*2/7+3/9*2/8*6/7 +6/9*5/8*4/7  = 0,452

Bharat B Sharma
Possible combinations = (all three odds); 1 odd two even; 2 odds one even, and all three even numbers
Probabilities of each  
P(3 odds)= (0.238095) 
P(2odd, 1even) = 0.535714
P(1 odds 2 even)= 0.21429
P(3 even) =  0.1191
Probability the total is odd = P all three odds + P 1 odd n 3 even = 0.2381+0.2143 = 0.4524.

Andrew Knox
Answer: Probability of the sum being odd is 228/504, or 0.45238095. 
There are four ways in which the sum of the three counters drawn from the bag can be odd, with associated probabilities:
1) odd - odd - odd, with probability 6/9 x 5/8 x 4/7 = 120/504
2) even - even - odd, with probability 3/9 x 2/8 x 6/7 = 36/504
3) even - odd - even, with probability 3/9 x 6/8 x 2/7 = 36/504
4) odd - even - even, with probability 6/9 x 3/8 x 2/7 = 36/504
So, probability the sum is odd, is the sum of these, i.e. 228/504 
(The reason why the chance is higher that the sum will be even (i.e. 276/404), in spite of there being twice as many odd numbers in the bag, is because the sum of two odd numbers is an even number, whereas the sum of two even numbers is not odd).

France Viellette
The total will be an odd number, if he picks 3 odd number counters, or 1 odd and 2 even counters. Therefore a probability of 120/504 + 108/504 = 228/504 or 19/42 or 45.24%.

Sergey Krasnozhon
P(ooo?eeo?eoe?oee)=6/9×5/8×4/7+3/9×2/8×6/7+3/9×6/8×2/7+6/9×3/8×2/7=38/84=0,45

Stephan Paischer
The odd total can be made up in two ways:
A – odd – odd – odd
The probability therefore is : 6/9 x 5/8 x 4/7 = 23,8%
(“the probability of odd in the first round is 6 out of 9, then 5 out of 8 (as one is already gone) and then 4 out of 7 (as two odd are already gone)”). The individual probabilities are multiplied using the mathematical “and” in stochastics.
Or
B – even – even – odd
The probability therefore is: 6/9 x 3/8 x 2/7 = 7,1 %
But this can also happen as even – odd – even or odd – even – even, for which the probability is each time the same.
So, you multiply by 3.
23,8 + 7,1*3 = 45,2%.

New teaser on Monday
 

On her first day at the office, rubber plant finance manager Elena introduces herself to her new team, all gathered around in a circle. They are evenly spaced and the seventh person in the circle is directly opposite the 18th. How many people are there at the introduction meeting altogether?

Answer: While some went round in circles, a good number of readers worked out that there were 22 people in the welcoming committee – or 23, depending on interpretation of Elena’s position. Very well done to: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; David Mann, key account manager, SPC Rubber Compounding, UK; Rohit Kalé, distribution strategy manager, AMN/V/B2C/DIS, Michelin North America Inc.: France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; Mehmet Koral, C&C Endüstriyel Dan??manl?k, E?itim ve Mümessillik Ltd, Göztepe, Istanbul, Turkey; Sergey Krasnozhon, SIBUR International GmbH, Vienna, Austria; Flemming Vorbeck, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; Jose Padron, material development specialist, Waterville TG Inc. Waterville, Québec, Canada; and everyone else who had a go.

Solutions

Just some of the neat working out, starting with Jose Padron’s splendid circle:

Mehmet Koral
Add the no. of team members plus Elena, the finance manager. 
There will be 10 people between them: the 8th, 9th, 10th, 11th, 12th, 13th, 14th, 15th, 16th, and 17th.
The two employees are directly opposite and all the others are evenly spaced, so there should also be 10 employees between them on the other side.
So, 10 between them on one side, 10 between them on the other, plus the 2 people themselves gives us 22 people together. Then we will add finance manager  Elena...

Sergey Krasnozhon
The 7th and the 18th are opposite, so  > 2. Drawing line between them (diameter for circle, 2 persons are directly opposite), will make 10 on one part of a circle > 10. Taking into account the second part   > 10. Total   > 22.

David Mann
Two solutions:
If the angle between 2 people is 360/x and 18 is opposite 7 then 11 people occupy 180 degrees, 22 people are in the circle. Alternatively, if you count around the circle then 1 must be sitting opposite 12, so there are 10 people between 1 and 12, 10 between 12 and 1 again making 22.

Michele Girardi 
You count up to 11 seats going from the 7th to the 18th; since you are half way, continuing until you are back to the 18th you count up to 22.

John Bowen
There are 22 people in the circle [8 to 18 inclusive ie 11 is half, so the full circle is 22] which is her team.  So including Elena there will be 23 people gathered altogether.

Andrew Knox
Total positions of sequential numbers around a circle (evenly spaced) is always the difference between opposites numbers times 2. So, in this case, (18-7) x 2 = 22, plus Elena in the centre is 23.

New teaser on Monday.

 

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

Answer: This slightly trickier variation on a previous teaser seemed to knock some of our Brainiacs out of shape. But not the following expert readers, who correctly answered 27: William A Alexander, associate professor, department of chemistry, drones research fellow, FedEx Institute of Technology, University of Memphis, Memphis, Tennessee, USA; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; 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. Well done to all and everyone else who had a go.

Solutions

There were some nice variations in approach to this question:

William A Alexander
Can be done by inspection of a simple spreadsheet, but there is a pattern. Four-digit squares run from 32 (1024) to 99 (9801) - 68 total possibilities to examine. If you look at the ones-place of the squares, the pattern (starting at 30) is 0,1,4,9,6,5,6,9,4,1,0; the pattern goes back and forth between 0 and 5.  This back and forth pattern continues for all perfect squares to infinity.
We're looking for anything with a ones-place digit of 1 or 9, which corresponds to 4 out of every ten choices. Thinking of 70 choices - 30 to 99 - yields 28 total squares, and we remove 1 (for 31) to get to a final answer of 27 total squares that differ from a multiple of ten by 1. 
The pattern is the key and explaining it takes way longer than doing it!

John Bowen
Smallest number is 33, largest is 99
For any set of numbers n1, n2, .... n9 where n=10s, the only ones with squares differing from a multiple of 10 are n1, n3, n7, and n9 which have squares ending 1,9,9,1.
So there are 3 in the 30s, and 4 in 40s, 50s, 60s, 70s, 80s, and 90s, so 27 in total.

Andrew Knox
If the question is interpreted as “which squares (of integers) with 4 digits differ by one from multiples of 10” i.e. end with a 1 or a 9, then the answer is 27. 
31 squared is 961 (too few digits)
32 squared is 1024 (okay)
99 squared is 9801 (okay)
100 squared is 10000 (too many)
So, in the 30 series only 3 work, the other 6 series (40 series through 90 series), 4 work, so the answer is 3 + (4 x 6) = 27.

France Veillette
There are 68 four-digit squares, from 322 to 992.  For the four-digit square to differ by one from a multiple of 10, it has to end by 1 or 9, therefore the number squared has to end with 1, 3, 7 or 9.  Between 32 and 99, there are 27 such numbers

David Mann
The four-digit range is from 1000 to 9999, so the roots of the square numbers must lie in the range of 32 to 100. To differ by 1 from these, the roots must end in 1,3,7 or 9.
So, the numbers that meet the criterion are; 33, 37, 39 41, 43, 47, 49 And so on until 99.
That makes 27 numbers in all.

Michele Girardi
The problem can be reformulated as "how many integers, squared, yield a number between 1000 and 9999, that end by 1 or 9?"
The target numbers must end by 1,3,7,9 and be between 32 and 99.
There are 4 numbers per decade and 7 decades, for a total of 28, but we must exclude 31 that is out of range, so the numbers are 27.

New teaser on Monday.