, 2022

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 g block and three 10 g blocks, and so Jasmine can
move the following total masses of blocks from Box B to Box A: 10 g,
20 g, 30 g, 50 g, 60 g, 70 g, and 80 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 g from Box B to Box A, then she must move blocks
whose total mass is $65 $\text{}$ g}+10 $\text{}$
g}=75 $\text{}$ g}$ from Box A to Box B so that Box A contains 65 g
less than it did originally and Box B contains 65 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 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 g
75 g

20 g
85 g

30 g
95 g

50 g
115 g

60 g
125 g

70 g
135 g

80 g
145 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 g block, one 20 g block,
and three 5 g blocks.

Is it possible for Jasmine to choose blocks from Box A whose total mass
is exactly 75 g?

Since the 100 g block is too heavy, and the remaining blocks have a
total mass that is less than 75 g, then it is not possible for Jasmine
to choose blocks from Box A whose total mass is exactly 75 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 g
75 g
No

20 g
85 g
No

30 g
95 g
No

50 g
115 g
Yes; one 100 g, three 5 g
4

60 g
125 g
Yes; one 100 g, one 20 g, one 5 g
3

70 g
135 g
Yes; one 100 g, one 20 g, three 5 g
5

80 g
145 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 $\text{}$ g})+1(20 $\text{}$ g})+
1(5 $\text{}$ g})=125 $\text{}$ g}$ from Box A to Box B and she moves
a total mass of $1(50 $\text{}$ g})+ 1(10 $\text{}$
g})=60 $\text{}$ g}$ from Box B to Box A.

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