This board state should be easily winning for us, the top player. We can either fill in black in row 1 - this tile is worth 9 points (7 for the adjacencies, 2 for the row bonus). Or we can fill in white in row 3 - this tile is worth 16 points (6 points for the adjacencies, 3 points for increasing the adjacencies of our already completed red tile, 7 points for the column bonus), although we will have to discard one of the whites, which should cost us about 2 points.
Still, just looking at this, it is clear that the white are significantly more valuable for us. At the same time, it is always important to look at the other side and see: What does our opponent get? Fortunately, this is not hard to calculate in this case. No matter what we take, our opponent will have something for row 2. However, 2 white are worth less for him (6 points after he places blue into row 1) than the 2 black (7 points), but the difference is negligible. There is also another problem for our opponent: We will try to group up as many blue tiles as possible to make him discard a lot, so we will look to take from the upper tray, not from the centre. Our opponent then faces a dilemma: Either he takes the 2 black (or the 2 white, depending on what we take), but allows us to converge all 5 blue. Or he takes 2 blue from the left tray to avoid costly discards, but gives us crucial white or black tiles in that process.
Even without calculating, it is already seems that taking white is correct, and no matter what our opponent does afterwards, it isn't favourable enough for him (although he should take the 2 black).
Still, one shouldn't assume but calculate, and also: This is a puzzle after all! So let's do exactly that, calculate it through.
To do that, I want to establish a convention first - how to do notation, similar to how chess moves are noted. I would propose this system and would appreciate all feedback:
- Trays are numbered 1 to 5 (the upper "Northern" tray is 1, and from there you go clockwise). The centre is 0.
- Colours are abbreviated as such: BL (Blue), Y (Yellow), R (Red), BK (Black), W (White)
- Rows 1 to 5 are numbered 1 to 5, obviously. The floor line is 0.
- Top player is Player A, bottom player is Player B.
You note a move by naming, in that order: 1) The tray you take from. 2) The colour you take. 3) The row you place it.
So, for example: 3-Y-2 simply means: You take yellow from tray 3 and put them in row 2. 0-BK-0 means: You take black from the centre and put them in your floor line. And so on.
Now that this cleared, let's look at all the options:
1) A: 1-W-3
(We take 2 white.)
- B: 0-BK-2 -> A: 5-Y-0 -> B: 0-BL-1 -> A: 0-R-4 =>
A: +32, B: +22, Δ: +10 for A
(Opponent takes the 2 black from the centre, we take 1 yellow from tray 5 and put it into our floorline. Alternatively, we could also take 1 red and put it into our floorline, it doesn't matter. Opponent takes all 5 blue for row 1, we put the remaining 3 red into row 4.)
- B: 5-BL-1 -> A: 0-BK-1 -> B: 0-Y-0 -> A: 0-BL-4 -> B: 0-R-5 =>
A: +41, B: +19, Δ: +22 for A
(Opponent takes the 2 blue from the left tray, everything converges in the centre. We take 2 black for row 1, opponent discards 1 yellow, we take 3 blue or red into row 4, opponent does the same with his row 5. We can see that this would not be an advisable move for the opponent.)
2) A: 1-BL-1
(We take 1 black from the upper tray).
- B: 0-W-2 -> A: 5-Y-0 -> B: 0-BL-1 -> A: 0-R-4 -> B: 0-BK-0 =>
A: +27, B: +19, Δ: +8 for A
(Opponent takes 2 white from the centre, we discard 1 yellow from the left tray and converge everything, our opponent takes the 5 blue and puts them into row 1, we put the 3 red into row 4, our opponent discards the remaining 1 black.)
As it's well established that the white tiles are insanely valuable for us, it is not necessary to consider our opponent not denying them and taking some blue tiles instead, this is obviously much worse.
Also note that it actually doesn't matter where we take the black tile from. If we go A: 0-BL-1, taking it from the centre instead, our opponent is still forced to take the two white, and the board state will be exactly the same afterwards.
Conclusion: We should take 2 white. This way, we will gain 10 points on our opponent (including all end-game bonuses for rows and columns). If we take 1 black, we will only gain 8 points instead. Interestingly enough, this is a much closer decision than it may seem at first glance!
It is hardly possible to calculate all the exact permutations in a short timespan, but we don't have to. The most important calculation: We know that the white tiles are worth 16 points for us, the black tiles only 9 points, that is a 7 point difference. It is also crucial to recognize: Our opponent will place 5 blue tiles in row 1 in any case. This is not so obvious, but with some experience, sheer gut feeling should tell you that our opponent will be forced to take either the two black or the white, depending on what we leave, because it gives him something for row 2 and denies us tons of points, even if this allows us to converge all blue tiles. This also means that our opponent will place black or white in row 2 no matter what. Also note that placing 5 blue in row 1 is +1 point (+5 for adjacencies, +2 for row bonus, -6 for discards), and placing 5 blue in row 5 is also +1 point, but without the row tiebreaker. This calculation becomes more tricky if our opponent has to discard more than 4 tiles, but this is very hard to compute within a game, so it's the easiest thing to just give him +1 for the blue tiles by default.
This simplifies the calculation. We have a 7 point difference between us taking white or black, but we will have to discard one more tile by taking white, so this difference shrinks to 5. Our opponent will also get black instead of white for his row 2, which is one more point for him, so the difference shrinks to 4. As it turns out, our opponent also discards one fewer tile if we take white, so that's the reason why difference shrinks 2 overall, but I wouldn't fault anyone for not seeing this during the game, because you actually have to calculate it through to know this.
If there is anything to learn from this:
- Try to simplify the calculation as much as possible. Try to disregard as many moves as possible because there is simply no time to consider all options. In a game, I would simply assume that my opponent can't afford to save himself some discards by taking blue from the left tray, which makes the possible decision tree fairly simple actually. Such assumptions can be misleading, but are often necessary depending on the time situation.
- If you have no time, trust your instinct. Your instinct will also get much better with more experience. I think most player's spontaneous decision would be to take white, because completing a column is better than completing a row after all! And as it turns out, this is correct, even though only by the slightest of margins.
- But if you do have the time, it is always better to indeed try to precisely calculate the options. If I had more than 60 seconds, ideally more than 90 seconds, I would do this for sure in this very situation. This also depends on your calculation prowess, of course. But I often see intermediate and even expert players rush their moves way too fast (that is also one big weakness of mine!). If you play against the absolute top players (which I do not belong to), you will see that it is fairly normal for a player to think more than a minute for a move if necessary.
Also, don't take my words for gospel. I may have miscalculated or overlooked something, so feel free to chime in with your own thoughts and maybe corrections!
