19

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 ...


19

First Way Second Way Third Way


8

The completed grid: Step-by-step solution: 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 ...


7

(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 Let's Now There is one ...


6

Solved sudoku (nothing fancy, just plain logical deduction): $$ \begin{array}{c|ccc|ccc|ccc|} & C1 & C2 & C3 \ & C4 & C5 & C6 \ & C7 & C8 & C9 \\ \hline R1 & 3 & 6 & 2 \ & 8 & 5 & 7 \ & 1 & 9 & 4 \\ R2 & 5 & 1 & 7 \ & 2 & 9 & 4 \ & 6 & ...


5

Probably not the smartest approach but works: (Confession: I did check with computer but only afterwards.) Taken together: The final result is therefore:


5

Surely must be enough, and also required. After 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 ...


4

I used integer linear programming (and, sorry, a computer): The solution is unique up to rotation and reflection.


2

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 ...


2

@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


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