To solve this, we need to examine more closely Grandpa's words when he says:
Two books of International stamps. One book of 75 cent stamps. And I got – 20 cent stamps, 50 cent stamps and also 55 cent stamps. That is it!
Knowing that Grandpa spent exactly \$50.00 and bought 13 books, what can we say for sure?
If we take his words at face value then we run ...
The completed grid:
We start by looking at the
We finally filled in a digit, just 80 more to go!
Lets look at the bottom box now,
Next we look at row 5 again.
Now we can make some progress in the middle:
We have completed two boxes, we are really getting somewhere!
To continue, we look at R3C7.
And with that, we have filled ...
(After I saw Rand's solution, it seemed to me that there ought to be a way of streamlining mine a bit. So I've done that. The original version of mine is preserved below in case anyone feels that the improved one is "polluted" by my having seen OP's own answer.)
The smallest possible $a+b$ is
First of all, notice that
There is one ...
must be enough, and also required.
white has two unavoidable captures coming up: on white's next move, the
Black can prevent only one of those moves, so
the first objective is completed.
Continuing from there, white will play
next. It threatens black's F-pawn, and also the queen's rook or some other piece in front of it. All those pieces ...
In the following, I've made quite a few assumptions, but to me they all seem somewhat sensible. Looks like there's a
chance we can use the first 7 blocks to build a shape with the outline given by OP, or the mirror image of it, which works just as well for the purposes of building a DT gun.
To see how this works, let's first improve OP's design by swapping ...
@PaulPanzer made a summary of existing answers here (including some they gave themselves).
I want to provide a clearer visualization for each of the existing solutions:
Source code for the above program