Each problem in the book gives a full board position, and tells you whether Black or White is to play. As this is how you might encounter an actual game, it is a very realistic approach. If you are stumped by a problem, you can refer to the hint beneath the diagram.
The book will help players from 10-kyu (most of whom will need the hints) all the way up to 4-5 dan. One drawback is that the book provides only one answer to each problem, and doesn't discuss other possible moves. But attempting all 501 problems will definitely help you develop a better understanding of how you should think during the opening and early middle game.
Problem 1: Black to play. First, just look at the board. Come up with a couple of possible solutions. For example, which area is larger? Is there a vital move that changes a weak group into a strong one, or vice versa? Better yet, can you make your group stronger at the expense of your opponent's group? Read the hint only after you have determined a move.
Hint: "Don't let your opponent link up his weak stones to his moyo!" Problem 2: White to play. Are there any weak groups? Are there any competing moyos?
Hint: "Push back the borders of your opponent's influence and expand our own!"
Solution 1: "Black 1 is the vital point. No other point is as crucial as this one. With the sequence to 7, Black prevents White from playing 3 and building a moyo in the center." This is the solution the book gives. What it doesn't mention is that Black will aim next at D or alternatively at A, followed by white B and black C, thereby isolating the marked white stones.
Solution 2: "White must play the diagonal move of 1. If Black were to play here, he would expand his influence on the left side while forcing the white stones to stretch along the bottom." This is the solution the book provides. However, that answer doesn't help if you thought white A was a vital spot. Shouldn't that be answered with black B? Actually, Black would play at 1, followed by white C, black D and white E. Now it becomes apparent the marked white stone is too close to white E. In other words, White's stones don't work well together. From that perspective, white 1 is an excellent move.
Solutions to last week's problems
Solution 2A: Black should play at 1 and, if White responds with 2, follow up with 3 to build up his moyo and keep up pressure on White. Next, he can attack the two lone white stones at the upper edge at A, or make territory with B, depending on the circumstances. Black 1 and 3 will probably give him the territory marked by the crosses, roughly 35 points. White, on the other hand, will probably retain the points shown by the triangles, about 11 points. The balance favors Black by 24 points, but he also spent an extra move. Using the QARTS system, you already know that a weak group (the white stones with only one eye) is valued at minus 10 points.
Solution 2B: Let's compare this with the situation in which White plays at 1. This will strengthen his group, making it immune to an attack (changing its QARTS status from minus 10 points to the value it has in terms of territory). After white 1, Black's corner has no base. Therefore he needs to play at 2 and 4 (normally this would be an endgame move, but Black is forced to play here at an early stage) to defend his corner. White needs to defend with 5, but this also builds up points in that area. It is quite likely White will get to play at C (because it also makes D sente), which is also part of the value of white 1 and should be calculated. White's territory is shown by the triangles, about 19 on the right side and 6 at the top, for a total of 25 points, whereas Black's territory is only the corner, worth less than 10 points. The balance favors White by roughly 15 points, but he has played two more moves compared to Solution 2A. However, the difference between these diagrams is about 40 points. Spending and extra two moves to make 40 points is not bad.
Solution 3A: Black should play at 1. What is this move worth? For one thing, it induces white 2, allowing Black to play 3 and build a huge territory in the lower right. Black's territory here is worth close to 35 points, while his upper right corner is worth about 14 points. White's stones are a little weak, but if he makes territory in that area, it is probably not more than about 6-7 points. Black is leading in this part of the board by more than 40 points. Furthermore, the white stones are not yet safe.
Solution 3B: What happens if White were to play at 1? Black should defend at 2. But now White's marked stones are strong, and he can use that strength to invade at 3. This may look dangerous, but as you can see, White can escape easily, laying waste to most of Black's territory here. Black's top right corner is worth about 10 points, while his lower right corner is now only worth about 15 points. On the other hand, White's marked stones have increased their value to nearly 15 points. The black stones around 4 and the white ones starting with 3 are both weak and provide no points. Black still leads on the right side by about 10 points, but this is only a shadow of his former lead of more than 40 points.
Conclusion: Strengthening your weak group or keeping your opponent's groups weak is very valuable. Because this changes or even reverses the balance of power, the rewards are often reaped in a different area of the board.