Tends to infinity

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Archive for the ‘Recreational’ Category

Sach ya Juth

Posted by tendstoinfinity on June 13, 2012

So once again I am here after a very long break for almost one and half year. Actually I was busy and more lethargic at times. Almost everyday I think I should share some interesting facts(there are so many!!!).I do not know the trustfulness of the sentence in the bracket. There is a very interesting riddle about persons telling  truth and false. Here it is……

Imagine that you are visiting an island on which there are only two kinds of people (other than yourself): satyavadi, who always tell the truth, and jutha, who always lie. There are two villages – one where all the satyavadis live, and another where all the juthas live. Although they live in separate villages, liars and truthers frequently roam about the island together and generally get along just fine. Talking to islanders is a bit difficult because they all observe the peculiar custom of not answering more than one question in a conversation and generally don’t elaborate on any statements they make. Another interesting feature of these islanders is that although outsiders can’t distinguish between satyavadis  and juthas by how they look.

1. Going to the Village

You are on the island and see a village on the road ahead of you, and you are not sure whether it is the satyavadi village or the jutha village. An islander, who may be a satyavadi or jutha, is standing on the side of the road. What one question do you ask her to find out if the village is the satyavadi village or the jutha village

2. Looking for the Ferry

You’ve decided to leave the island and are trying to find the ferry that will take you back to the mainland. There is a fork in the road that splits off in two directions. Two islanders, Anand and Bhaskar, are standing at the fork. Anand and Bhaskar are from different villages; you don’t know who is from the truther village and who is from the liar village, and Anand and Bhaskar won’t answer questions about their villages. What question do you ask one of them to find out how to get to the ferry?

3. On the Ferry

It’s a slow day, and you are the only passenger on the Ferry: it is just you and the captain. As it pulls out into the harbour you realize that you might have boarded the wrong ferry – is this really the boat that is going to the mainland? You can ask the captain, an islander himself, one question to find out.

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Posted in Reasoning, Recreational | 2 Comments »

salesman problem

Posted by tendstoinfinity on February 10, 2011

pichak@scientist.com

Perhaps you have remembered Dr. A N Mishra, KV Jorhat one of the teachers blessed with splendid presence of mind. He has asked me a very beautiful question.

One salesman used to carry 40 different weights 1kg,2kg,….40kg for weighing articles upto 40kgs, until somebody suggested why doesn’t he use less number of weights so that he can weigh all the kgs upto 40kg in one turn. Now can you help the salesman to choose minimum number of weights in the following cases

i. weights can be placed on either of the  pans of the balance.(this is Dr. Mishra’s question)

ii. weights can be placed only on left pan and object on right pan.( this is my variation)

before presenting the solutions directlyI want to see the Result: All the naturals up to 2^{n+1}-1 can  be obtained by adding all or some of the element of the set  {1=20,21,22,23,…2n}    only once. (Why?) For example by adding some or all of the set only once you can get all naturals upto 7 using all or some {1,2,4}. You can see following table for illustaration

No of 4 No of 2 No of 1 sum
0 0 1 1
0 1 0 2
0 1 1 3
1 0 0 4
*1 0 1 5*
#1 1 0 6
1 1 1 7

Are you able to see that in the above table (no of 4,no of 2 and no of 1) form the binary ( number system of base 2)representation of the respective sum. *Just as (101)2 =(5)10 . Now binary representation of 8 to 15 =24-1 contains 4digits. So the sum of any natural number from 1 to 15 can be represented as a sum of all or some of {20=1, 2, 4, 8=23} only once. I think my variation i.e (ii) can now be  solved. Because you can only use one pan to place weights so addition only should be used so 6 weights of {1,2,4,8,16,32} will be necessary for  25-1<40<26-1. In fact the salesman can weigh any weights upto 61kg in one turn using these weights.

Another approach of   thinking

There are two places for the weight units either in the balance pan or in bag of salesman. Let 0 represents the bag and 1 represents pan so weighing a weight of 6kg we have representation 110# means 4kg, 2kg on the pan and  1 kg in the bag.

Before we turn to Dr. Mishra’s problem we want to see the a wonderful property of ternary system(number system of base 3) : every natural can be represented as the sum or difference of two numbers whose ternary representation contain only 0’s and 1’s or otherwise every natural can be expressed as the sum or differences of numbers of the form 3n(n is a natural no.). If the natural number is a power of 3(of the form 3n) the prove is obvious. Now for the other case see the following example

(4)10=(11)3=(11)3-03 = 31+30

(5)10= (12)3

= 1.31+2.30

= 1.3+(3-1)30

= 1.3+3-1

=2.3-1

=(3-1)3-1

=32-31-30

= (100)3-(10)3-10

I hope you can generalize the facts.

So every natural number upto  \frac{3^{n+1}-1}{2}can be expressed by adding or subtracting all or some elements of the set {1=30,31,32,…3n}

So every number less than 40 can be expressed as sum or difference of all or some of {1,3,9,27}. These are weights required by the salesman.

For base 4, Similarly every natural number can be expressed by sum or difference of the powers of 4 taking any specific element not more than twice.

5 = 41+40

6 = 4+1+1 (1 =40)

7 = 16 – 4 – 4 – 1

53 = 64 – 16 + 4 +1

Its proof is  so simple. But giving a little time you can prove it.

Hopefully in some post I will discuss some more interesting fact of the bases of number system.

Posted in Recreational, Uncategorized | 3 Comments »