ERJ Brainteaser: April

Sean has secured his suitcase with two combination locks, each having a five-digit code. For the codes, he decided to use the smallest and largest numbers that met the following conditions:

Each digit of the number was a prime digit,

Each successive pair of digits formed a two-digit number that was not a prime number,

Each of the prime digits had to appear at least once in the five-digit number.

What were the codes for Sean’s combination locks?

Answer: Well done to our top code-breakers: John Bowen, consultant, Bromsgrove, UK; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Bharat B Sharma, technical director, Rajsha Chemicals Pvt. Ltd, (TWC Group), Vadodara (Guj) India; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; and everyone else who had a go.

SOLUTIONS

John Bowen

There are only 4 prime digits [1 by definition is not a prime]

2, 3, 5, 7.

Smallest must start with 3 [23 is prime] so is 32527

Largest must also start with 3, so largest is 35752

[each pair of digits to be non-prime]

Kamila Staszewska

Allowed digits: 2, 3, 5, 7

Allowed pairs: 22, 32, 52, 72, 33, 25, 35, 55, 75, 27, 57, 77

Smallest 5-digit combination: 32257 & Largest: 35772

Sudi Sudarshan

The prime digits are 2, 3, 5, 7

Any 5-digit code that has only prime digits and all the prime digits will have one prime digit repeating.

Since all pairs of consecutive digits have to be non-prime, the following consecutive digits are invalid as the two-digit numbers they form are not prime: 23, 37, 53, 73. So 3 can only be in the first place or in the 2nd place only if preceded by a 3

5-digit codes that meet the above criteria:

32257, 32275, 32527, 32557, 32572, 32575, 32577, 32725, 32752, 32755, 32757, 32775, 33257, 33275, 33527, 33572, 33725, 33752, 35227, 35272, 35257, 35275, 35277, 35527, 35572, 35722, 35725, 35727, 35752, 35772

Smallest number is 32257 and largest number is 35772

Bharat B Sharma

Prime numbers between 1 to 100 are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Prime numbers in single digit are 2 3 5 7

Each successive pair of digits formed a two-digit number to be non prime that means the combination numbers 23,53,73 can not come any where in middle of 5 digits code (being prime number).

Hence the number need to start with 3.  Utilizing remaining three number to form lowest 4 digits numbers - which can be 2257 (lowest) and 7752 (highest) (as we have to use all 4 prime digits in the number and we have chosen 3 as the first digit).

The two codes, therefore could be 32257 and 37752 

ALTERNATIVELY…

Andrew Knox

The number we are looking for is: 23577

We are looking for five-digit numbers that meet the following three conditions:

We will generate all 5-digit permutations using {2, 3, 5, 7}, with repetition allowed.

That's 1024 total combinations.

From those 1024:

Keep only numbers that use all of 2, 3, 5, 7. For each number, check if the pairs formed by its digits are not prime.

After checking all possibilities, the five-digit code is:

23577

These numbers:

Use only digits 2, 3, 5, 7.

Contain all four digits.

None of their adjacent digit pairs (like 23, 35, 57, 77, etc.) form a prime two-digit number.

Amparo Botella

The number we are looking for is: 23577

We are looking for five-digit numbers that meet the following three conditions:

We will generate all 5-digit permutations using {2, 3, 5, 7}, with repetition allowed.

That's 1024 total combinations.

From those 1024:

Keep only numbers that use all of 2, 3, 5, 7. For each number, check if the pairs formed by its digits are not prime.

After checking all possibilities, the five-digit code is:

23577

These numbers:

Use only digits 2, 3, 5, 7.

Contain all four digits.

None of their adjacent digit pairs (like 23, 35, 57, 77, etc.) form a prime two-digit number.

If it takes Tim 1 hour and 40 minutes to fell a tree, how long will the same job take if he is joined by Terry and Tom working respectively 10% and 20% faster?

Answer: A select group of our top Brainiacs made short work (see Solutions) of this EUDR-compliant question: Well done to: Bharat B Sharma, technical director, TWC Group (Rajsha Chemicals Pvt. Ltd), Vadodara (Guj) India; John Bowen, consultant, Bromsgrove, UK; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; and everyone else who had a go. 

SOLUTIONS

Bharat B Sharma

Let’s say, the speed of cutting tree (Choppinmg speed) of Tim is x/minute

Total chopping required in1hour40 Minutes (time taken by Tim is 100 minutes) = 100 xChoppings.

Chopping Speed of Terry = 1.1x/minute (10% faster than Tim)

Chopping Speed of Tom = 1.2x/minute (20% faster than Tim)

If all three work together, the total speed will be x+1.1x+1.2x = 3.3x/minute

Total time that will be required = 100x / 3.3x  minutes = 100/3.3 minutes = 30.303 minutes!

Answer-  30.303 Minutes

John Bowen

Let Tim's work rate = 1.0, so to fell a tree takes 100 man minutes.

Total work rate when all 3 are working together = 3.3, so time to do 100 man minutes of work is now 100/3.3 minutes

 = 30.30 minutes or 3minutes 18 seconds

Kamila Staszewska

Tim cuts 1 tree in 1h40min=100min; 1/100=0.01 tree/min

Terry: +10% = 1.1 tree per 100min=0.011 tree/min

Tom: +20% = 1.2 tree per 100min=0.012 tree/min

Workers combined = (1+1.1+1.2)/100= 3.3/100=0.033 tree/min

0.033 tree - 1 min

1 tree - x min

x= 1/0.033=30.3min=30min18s

Sudi Sudarshan

Solution:

Tim's speed of felling a tree  = 1/100 trees per minute

Terry's speed of felling a tree  = (1/100)*1.1

Tom's speed of felling a tree = (1/100)*1.2

Combined speed of the three men for felling a tree  = (1/100)+ (1/100)*1.1 + (1/100)*1.2   = (1/100)* (1+1.1+1.2) = 3.3/100 trees per minute

Time taken by the three men working together to fell a tree = 1/(3.3/100) = 100/3.3 = 30.3030...

So, it takes 30.30 minutes for the three men together to fell a tree.

Amparo Botella

The time to cut 1 tree is 30 minutes and 18 seconds.

Tim takes 1 hour and 40 minutes, which is 100 minutes, to fell a tree.

So his work rate is 1 tree100 minutes=0.01 trees/minute

Terry is 10% faster than Tim, so his rate=0.01×1.10=0.011 trees/minute

Add all their rates together:

Total rate=0.01+0.011+0.012=0.033 trees/minute

if in 1 min. we cut 0.033 trees then 1 tree will be cut in: 1/0.033= 30.3 minutes

That’s approximately 30 minutes and 18 seconds.

New teaser on Monday