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You can place weights on both side of weighing balance and you need to measure all weights between 1 and 1000. For example if you have weights 1 and 3,now you can measure 1,3 and 4 like earlier case, and also you can measure 2,by placing 3 on one side and 1 on the side which contain the substance to be weighed.
So question again is how many minimum weights and of what denominations you need to measure all weights from 1kg to 1000kg.
We require only......Click here to know!
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What was the task given?
Just for a moment, let's assume we have to weigh 1 to 30 Kg. Now you can weigh all those with
1, 2, 4, 6, 8, 10..........30
1 2, (2+1), 4, (4+1),6, (6+1), 8,(9+1),10..(29+1),30 .......For middle weights.
Now 6 can be weighed as 2 + 4 & 10 can be weighed as 2 + 8. We can eliminate those. That means we require only
1, 2, 4, 8, 16,
So we need weights of powers of 2.
Now if subtraction is allowed then,we require
1, 3, 6, 9, 12, 15, 18,...............30
For all weights ,
1, (3-1),3, (3+1), (6-1), 6, (6+1), (9-1),9, 12, 15,............30
But 6 can be weighed as 9-3 , 12 as 9+3, 15 as 27-(9+3). So we can eliminate 6,12,15... This leaves only
1, 3, 9, 27,
In short, we need power of 3 only.
For the given problem we need to weight 1 to 1000 Kg with subtraction allowed. So the maximum power of 3 that is less than 1000 is 7. To conclude, we require only 7 weights as below.
1,3,9,27,81,243,729
There are 100 people standing in a circle in an order 1 to 100. No. 1 has a sword. He kills the next person (i.e. No. 2) and gives the sword to the next (i.e. No. 3). All people do the same until only 1 survives. Which number survives at the last?
Find who survived?
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What was the problem?
First let's make it very simple by who are surviving after each round.
Round 1 : 1,3,5,7,9,11,13,15..........87,89,91,93,97,99
Round 2 : 1,5,9,13,17.........89,93,97
Round 3 : 1,9,17,25,33,41,49,57,65,73,81,89,97
Round 4 : 9,25,41,57,73,89
Round 5 : 9,41,73
Round 6 : 9, 73
At round 5, 9 kills 41 & passes sword to 73. So 73 kills 9 & survives.
Round 7 : 73
Now let's analyze how this happens & trick to get answer at the quickest.
Just for a moment let's assume there were 16 standing in circle.
Now after each round survived people are,
Round 1 : 1,3,5,7,9,11,13,15
Round 2 : 1,5,9,13
Round 3 : 1,9
Round 4 : 1
After round 2, when 9 kills 13 & passes sword to 1.
And in round 4, 1 kills 9. So 1 forms pair with other in every round.
Now imagine there were 17 people in circle.
