When my boss sent me to stone age.

I am writing this entry with assumption, my boss don't kill me after reading this post.

Once my super boss asked me to go to stone age to help a new shopkeeper who wanted to become guru in grains selling business. Being smart chap, he did some market analysis on selling of grains. He then found out the customers are purchasing grains weighing any integral value from 1 to 40. Now he went to kilomaker and requested him for few weights. But Kilomaker said, he can prepare only four weights for him.

Now, how should i help shopkeeper to choose the four weights, that can be used to measure any weight from 1 to 40 in one go?

Now i can come back only if i finish my task.



Solutions:

Possiblites using mathematical operation(summation and substraction):

1) a
2) b
3) c
4) d


5) a + b
6) a + c
7) a + d
8) b + c
9) b + d
10) c + d


11) a + b + c
12) a + b + d
13) a + c + d
14) b + c + d


15) a + b + c + d


16) a - b
17) a - c
18) a - d
19) b - c
20) b - d
21) c - d


22) a + b - c
23) a + b - d
24) a + c - b
25) a + c - d
26) a + d - b
27) a + d - c
28) b + c - a
29) b + c - d
30) b + d - a
31) b + d - c
32) c + d - a
33) c + d - b


34) a + b - c - d
35) a + c - b - d
36) a + d - b - c


37) a + b + c - d
38) a + b + d - c
39) a + c + d - b
40) b + c + d - a





Final Answer:

1, 3, 9, 27



Brief explaination:

1, 3 , 9 can uniquely represents 1 to 13.
So it can be used to get number from (14 to 26) if substracted through 27
and number(28 + 40) if added to 27.




Detailed explaination:

1 = 1
2 = 3 - 1
3 = 3
4 = 3 + 1
5 = 9 - 3 - 1
6 = 9 - 3
7 = 9 - 3 + 1
8 = 9 - 1
9 = 9
10 = 9 + 1
11 = 9 + 3 - 1
12 = 9 + 3
13 = 9 + 3 + 1
14 = 27 - 9 - 3 - 1
15 = 27 - 9 - 3
16 = 27 - 9 - 3 + 1
17 = 27 - 9 - 1
18 = 27 - 9
19 = 27 - 9 + 1
20 = 27 - 9 + 3 - 1
21 = 27 - 9 + 3
22 = 27 - 9 + 3 + 1
23 = 27 - 3 - 1
24 = 27 - 3
25 = 27 - 3 + 1
26 = 27 - 1
27 = 27
28 = 27 + 1
29 = 27 + 3 - 1
30 = 27 + 3
31 = 27 + 3 + 1
32 = 27 + 9 - 3 - 1
33 = 27 + 9 - 3
34 = 27 + 9 - 3 + 1
35 = 27 + 9 - 1
36 = 27 + 9
37 = 27 + 9 + 1
38 = 27 + 9 + 3 - 1
39 = 27 + 9 + 3
40 = 27 + 9 + 3 + 1





Moving towards rays in dark



How to come on these number ?


Step 1: We need to measure 1, conclusion
a. One of number should be 1. Or,
b. Two number should be consecutive.

Step 2: We need to measure 2,
a. Two number difference should be 2.
b. Second number should be 2.

Step 3: We need to measure 3:
a. The number should be itself 3.
b .Two number difference 3.
c. Addition of 1 and 2 as 3. This will voilate our uniqure constraints ( 2 - 1 = 1 i.e a - b = b) .

So we can have either 1 or 2.

So, one of the possible two number are 1 and 3. And these uniquely represent 1, 2, 3, 4 and it will
also helps us to represent 40, 39 , 38 , 37(to represents +4, -4 number from a given number.)


Going ahead on this line,

we are left with c + d = 36.

Now to represents 1 to uniquely
third number should be 9 or difference of other two number i.e c and d should be 9.

let assume second case,

c + (c + 9) = 36
=> 2c = 27

which will give us decimal value so ruled out. So third number is 9 and which makes 4rth as 27.




~ njoy
Anand