Question

Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken has to prepare n cups of cubes, where n is as small as possible, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?

• A. 22
• B. 64
• C. 81
• D. 124

Answer 1

The correct option is B

64

This question is entirely based on the powers of 2 less than 127.

Thus the answer is $2^6=64$ cubes in the cup with the most sugar.

How is that? Let us look at this important concept using the unitary approach:
Take the number 5.
In other words the question is - how many numbers would you need so that all the numbers from 1 to 5 are possible using addition.
The powers of 2 less than 5 are $2^0=1,2^1=2$ and $2^2=4$.
1= 1
2= 2
3= 1+2
4=4
5=4+1
Thus, we see that with powers of 2 we can easily write all the numbers below the highest power of 2 taken.
In case we take up to $2^2$, then we can write all values upto $2^3-1$ or 7.
$2^0=1$
$2^1=2$
$2^1+2^0=3$
$2^2=4$
$2^2+2^0=5$
$2^2+2^1=6$
$2^2+2^1+2^0=7$

Hence with 3 numbers you can write upto 7.

Hence to add up to 127 you will have to consider the powers of 2 till $2^6$.

So the cups must contain 1, 2, 4, 8, 16, 32, and 64 cubes of sugar, with 64 being the most number of cubes in one cup.