Question 1

There is stone weighing 125 kg on the top of a hill. One day, the stone suddenly due to some reason, starts sliding from the hill and gets broken in pieces. While sliding it gets broken down in several small stones , each having a integer weight, and no loss of mass considered. Suppose it gets split into n small stones.
Now weight of the stones are quite interesting. Tom standing on the top of the hill realise that by doing addition and subtraction of the weights of the stones, using weight of each stone exactly once , he was able to make most of the numbers from 1 to 125.
But still he could not form some m numbers ranging from 1 to 125. But he realises that they are such that value of m+n is minimum.
Can you also tell the minimum value of m+n?

Comments on Question 1

Solution

Answer: 2)7
Let us split 125 as 121+4.
121= 1+3+9+27+81
4=1+3
Now one special property of powers of 3 is that they can added , subtracted to get every number from 1 to their sum. Like 1,3,9 can be added , subtracted once to make every number from 1 to 1+3+9.
This can be seen as
3^0+ 3^1 ... +3^n= {[3^(n+1)] -1 }/2
So 1, 3,9,27,81 make every number from 1 to 121. Now for next 4, we again can assume 2 stones of weight of 1 and 3. So we can make any number from 1 to 125 now. So m becomes 0 and n is 7. Also powers of 3 gives the splitting in minimum numbers so that we can achieve the given property.
If we take pieces to be of 1,3,9,27,82,4, then we cannot make 122,123,124; so m is 3 and n is 6. So m+n becomes 9. Also as 1,3,9,27,81 are the numbers which by adding and subtracting can give maximum sum and minimum number of pieces. So for next 4 we will surely ge tone more stone and also will leave some cannot form numbers so m+n must be greater than 6.