# Expeditions

### by Joseph DeVincentis

ANSWER: OSCITANCES

From Clue 2, there were no cards in the discard piles at the end. The rules state that the game ends when the last card is drawn from the draw pile, so there were no cards there either. Thus, save for the 16 cards remaining in the players' hands, they were all in play. We have no information about which cards are in which remaining hand, except that the red 3 is in the last player's hand, and we don't know who that is! So treat each card as being in one of three places (played by Alphonse, played by Betty, or unplayed), and use an elimination grid like this (in which x stands for an investment card and T for 10):

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |            |            |            |            |            |
Alphonse |            |            |            |            |            |
hand     |            |            |            |            |            |total 16

```

We immediately know that Alphonse played no 6s, Betty played no 2s (Clue 6), Betty played the green 6 and Alphonse played a white investment card (Clue 8), nobody played the red 3 but all other odd-numbered cards were played (Clue 9), and all 10s were played (Clue 12). Exactly 6 expedition cards were played (Clue 3), so 9 of them went unplayed. Besides these 9 expedition cards and the red 3, and one 2 (Clue 6), there were five more unplayed cards which were 4s, 6s, and 8s. Clue 6 also implies that at least two of the 6s were unplayed.

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |   -        |   -        |-  -        |   -   6    |   --       |x
Alphonse |       -    |       -    |x      -    |       -    |    -  -    |xxxx
hand     |    - - - --|    - - - --|-   - - - --|    - --- --|    3 - - --|xxxxxxxxx266
& three of
44466888

```

Since Alphonse only played two yellow numbers (Clue 5), he got a negative score there, and so did not play a yellow investment card (Clue 3). Alphonse played two blue numbers matching yellow numbers he played (Clue 5) and four blue numbers matching yellow numbers Betty played, and only 6 cards in all (Clue 2), so he played no blue investment cards. Betty played only one investment card (Clue 3) so at least two yellow and two blue investment cards went unplayed. Alphonse played at most two white investment cards, and then only if Betty's investment card was green (Clue 8), so at least one white investment card went unplayed. Also since Betty did not play the green 2, Alphonse could not have played the white 2 (Clue 8), so he must have played all the other ones and the white 2 was unplayed (Clue 6).

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |-- -        |-- -        |-- -        |   -   6    |   --       |YYYY x
Alphonse |---2   -    |---2   -    |x- -   -    |   2   -    |   2-  -    |YBBBBB xxxx
hand     |xx -- - - --|xx -- - - --|-x 2- - - --|   -- --- --|   -3 - - --|xxxx66 & three
of 44466888

```

Since Alphonse played a white investment card, he made a positive score in white (Clue 3) which means he played at least three white numbers. Since white is the only color where he did not play a 2, by Clue 11 he played no even numbers in white and got an odd score. This means he must have played exactly two investment cards (to make the multiplier odd) and exactly three numbers (to make the number sum odd) and the only three odd numbers large enough are 5, 7, and 9. So Betty played exactly 5, 6, 7, 9, and one investment card in green (Clue 8) and no other investment cards (Clue 4). We are also told Betty played at least four yellow numbers (Clue 10), so she played exactly three cards in blue, white, and red (Clue 4). By elimination, Betty played the white 3 and 10 and Alphonse played the green 3 and 10, and all the yellow and blue investment cards were unplayed.

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |----        |----        |----3 - - -T|x-----567-9-|-----       |YYYYBBBWRRR
Alphonse |---2   -    |---2   -    |x-x---5-7-9-|-  23 --- -T|   2-  -    |YBBBBB xxx
hand     |xxx-- - - --|xxx-- - - --|-x-2- - - --|-  -- --- --|   -3 - - --|xx66 & three
of 44466888

```

Since Betty played three blue numbers and Alphonse played six blue numbers, all the blue numbers were played. So Betty played the blue 6, and since her three numbers here are consecutive (Clue 4), Alphonse played the blue 3, 9, and 10. Betty played yellow numbers matching four of Alphonse's six blue numbers, and not the 2, so the candidates are 3, 9, 10, and two of 4, 5, 7, 8. The minimum value of the four cards is thus 3+4+5+9=21, so Betty made a positive score in yellow. We also know she did so in green, but by clue 7, she did not do so in two non-red expeditions, so these must be blue and white. This means she did not play the white 8 (which is instead unplayed) and her consecutive blue cards were not 6,7,8, so she played the blue 5 and Alphonse played the blue 8. By the last part of Clue 4, Betty did not play 5,6,7 in blue since she has that sequence in green. So she played 4,5,6 and Alphonse played the blue 7 as well.

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |----        |-----456----|----3 - ---T|x-----567-9-|-----       |YYYYWRRR
Alphonse |---2   -    |---23---789T|x-x---5-7-9-|-  23 --- -T|   2-  -    |Y xxx
hand     |xxx-- - - --|xxx---------|-x-2- - -8--|-  -- --- --|   -3 - - --|xx66 & two
of 44688

```

Betty played three consecutive red cards (Clue 4) which do not include the 3, and are not 4,5,6 or 5,6,7 (Clue 4), so they include the 8. This means Alphonse's red cards add to at most 2+4+5+9+10=30, or a score of 10. Since Alphonse's red expedition had the highest score of any by either player, and he scored 2+3+7+8+9+10-20=19 in blue, he must have played an investment card in red. It doesn't work for him to have played two investment cards, since he played at most 6 red cards (Clue 2) and removing one it reduces the card score to at most a score of 6, which would yield a total of 18. Having three investment cards makes his score even lower. To beat the 19 points in blue, he had to play exactly 2, 4, 5, 9, 10 and an investment card, leaving 6, 7, 8 played by Betty. Since Betty has played three 6s, by Clue 6 she did not play any more; the white and yellow ones were unplayed, and her last white card was the 4. By elimination, Alphonse's last two investment cards were green. To get a positive score in green he must have played the 8. Playing the green 4 also would give him 21 points in green, so that card was unplayed.

```          yellow       blue         white        green        red          partial info
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |----   -    |-----456----|----34-----T|x-----567-9-|-------678--|YYYY
Alphonse |---2   -    |---23---789T|x-x---5-7-9-|-xx23----8-T|x--2-45---9T|Y
hand     |xxx-- -6- --|xxx---------|-x-2---6-8--|-----4------|-xx-3-------|one of 48

```

Alphonse's other yellow card (besides the 2) was a number he also played in blue and green (Clue 5) but not in red or white (since it would give him 4 of them, in violation of Clue 6). That makes it the 3 or 8. So Betty played the yellow 7, 9, and 10. Betty played three consecutive numbers in yellow (Clue 4). If Alphonse played the yellow 8, Betty's consecutive yellows must be 3,4,5. But that leaves only 15 total unplayed cards. So Alphonse played the yellow 3 (and no other yellows besides the 2), and Betty played the yellow 8. By elimination, Betty played the yellow 5, and the yellow 4 was unplayed to make 16 unplayed cards. So the final layout is:

```          yellow       blue         white        green        red
xxx23456789T xxx23456789T xxx23456789T xxx23456789T xxx23456789T
Betty    |------5-789T|-----456----|----34-----T|x-----567-9-|-------678--|
Alphonse |---23-------|---23---789T|x-x---5-7-9-|-xx23----8-T|x--2-45---9T|
hand     |xxx--4-6----|xxx---------|-x-2---6-8--|-----4------|-xx-3-------|
```