Box A contains one $100 \mathrm{~g}$ block, one $20 \mathrm{~g}$ block and three $5 \mathrm{~g}$ blocks. Box B contains one $50 \mathrm{~g}$ block and three $10 \mathrm{~g}$ blocks. Jasmine moves some of the blocks from Box A to Box B and some of the blocks from Box B to Box A. After these moves, Box A contains $65 \mathrm{~g}$ less than it originally did and Box B contains $65 \mathrm{~g}$ more. What is the fewest number of blocks that Jasmine could have moved from Box A to Box B?
(A) 3
(B) 4
(C) 2
(D) 5
(E) 1
We begin by recognizing that moving a block from Box A to Box B, and then moving the same block from Box B back to Box A has does not change the total mass in each box.

Since such a pair of moves increases the number of blocks that Jasmine moves from Box A to Box B, then all such pairs of moves will not occur in counting the fewest number of blocks that Jasmine could have moved from Box A to Box B.

(Similarly, moving a block from Box B to Box A, and then moving that same block from Box A back to Box $B$ is a pair of moves that will not occur.)

That is, once a block has been moved from one box to another, that block will not be moved in the opposite direction between the two boxes.

Next, we consider what total masses of blocks can be moved from Box B to Box A.

Box B contains one $50 \mathrm{~g}$ block and three $10 \mathrm{~g}$ blocks, and so Jasmine can move the following total masses of blocks from Box B to Box A: $10 \mathrm{~g}, 20 \mathrm{~g}, 30 \mathrm{~g}, 50 \mathrm{~g}, 60 \mathrm{~g}, 70 \mathrm{~g}$, and $80 \mathrm{~g}$. (Can you see how to get each of these masses using the blocks in Box B and why no other masses are possible?)

If Jasmine moves $10 \mathrm{~g}$ from Box B to Box A, then she must move blocks whose total mass is $65 \mathrm{~g}+10 \mathrm{~g}=75 \mathrm{~g}$ from Box A to Box B so that Box A contains $65 \mathrm{~g}$ less than it did originally and Box B contains $65 \mathrm{~g}$ more than it did originally.

For each of the other masses that may be moved from Box B to Box A, we summarize the total mass of blocks that Jasmine must move from Box A to Box B. Note that in each case, the mass moved from Box A to Box B must be $65 \mathrm{~g}$ greater than the mass moved from Box B to Box A.

| Mass moved from  Box B to Box A | Mass moved from  Box A to Box B |
| :---: | :---: |
| $10 \mathrm{~g}$ | $75 \mathrm{~g}$ |
| $20 \mathrm{~g}$ | $85 \mathrm{~g}$ |
| $30 \mathrm{~g}$ | $95 \mathrm{~g}$ |
| $50 \mathrm{~g}$ | $115 \mathrm{~g}$ |
| $60 \mathrm{~g}$ | $125 \mathrm{~g}$ |
| $70 \mathrm{~g}$ | $135 \mathrm{~g}$ |
| $80 \mathrm{~g}$ | $145 \mathrm{~g}$ |

Next, we determine if it is possible for Jasmine to choose blocks from Box A whose total mass is given in the second column in the table above.

Recall that Box A originally contains one $100 \mathrm{~g}$ block, one $20 \mathrm{~g}$ block, and three $5 \mathrm{~g}$ blocks. Is it possible for Jasmine to choose blocks from Box A whose total mass is exactly $75 \mathrm{~g}$ ?

Since the $100 \mathrm{~g}$ block is too heavy, and the remaining blocks have a total mass that is less than $75 \mathrm{~g}$, then it is not possible for Jasmine to choose blocks from Box A whose total mass is exactly $75 \mathrm{~g}$.

In the table below, we determine which of the exact total masses Jasmine is able to move from Box A to Box B.

For those masses which are possible, we list the number of blocks that Jasmine must move from Box A to Box B.

| Total mass  moved from  Box B to Box A | Total mass  moved from  Box A to Box B | Is it possible  to move this mass  from Box A to Box B? | Number of blocks  moved from  Box A to Box B |
| :---: | :---: | :---: | :---: |
| $10 \mathrm{~g}$ | $75 \mathrm{~g}$ | No |  |
| $20 \mathrm{~g}$ | $85 \mathrm{~g}$ | No |  |
| $30 \mathrm{~g}$ | $95 \mathrm{~g}$ | No | 4 |
| $50 \mathrm{~g}$ | $115 \mathrm{~g}$ | Yes; one $100 \mathrm{~g}$, three $5 \mathrm{~g}$ | 3 |
| $60 \mathrm{~g}$ | $125 \mathrm{~g}$ | Yes; one $100 \mathrm{~g}$, one $20 \mathrm{~g}$, one $5 \mathrm{~g}$ | 5 |
| $70 \mathrm{~g}$ | $135 \mathrm{~g}$ | Yes; one $100 \mathrm{~g}$, one $20 \mathrm{~g}$, three $5 \mathrm{~g}$ |  |
| $80 \mathrm{~g}$ | $145 \mathrm{~g}$ | No |  |

From the table above, the fewest number of blocks that Jasmine could have moved from Box A to Box B is 3. In this case, Jasmine moves a total mass of $1(100 \mathrm{~g})+1(20 \mathrm{~g})+1(5 \mathrm{~g})=125 \mathrm{~g}$ from Box A to Box B and she moves a total mass of $1(50 \mathrm{~g})+1(10 \mathrm{~g})=60 \mathrm{~g}$ from Box B to Box A.

This confirms that Box B contains $125 \mathrm{~g}-60 \mathrm{~g}=65 \mathrm{~g}$ more than it did originally (and hence Box A contains $65 \mathrm{~g}$ less), as required.

ANSWER: (A)