PSE Advent Calendar 2021 (Day 11): What Child – Er, Game Is This?

This puzzle is part of the Puzzling StackExchange Advent Calendar 2021. The accepted answer to this question will be awarded a bounty worth 50 reputation.

< Previous Door Next Door >

I. The story

The snow was blowing, the hot cocoa was warm, and I had just settled down for the night with my friend Alice's family to play a holiday game or two before going home to my own family for Christmas. To my surprise, however, instead of us all playing a friendly game of Codenames, Taboo or Yahtzee, I was let in on a holiday secret.

"Every year," Alice explained, "we play a secret holiday game. It's a Christmas-themed variation of a well-known game that anyone can play, but nobody outside of our family is allowed to know the rules we use, how to play, or how to determine the winner and loser. It's our Christmas secret."

"We'll play one round of the game for you, and give you a chance to learn it from listening to us play," her brother, Bob, added. "Both of us won't necessarily play perfectly, but we'll be playing strategically to try and stop the other from winning. You don't need any other information except for the words we say out loud, although a piece of paper and a pencil could help keep track. Then we'll play a second round of the game, and if you can figure out what the game is from there, you'll get to be part of the secret from now on."

"You're on," I said, confident in my puzzling abilities.

II. The game

The first round of the game played out as follows.

1. Alice: "Jingle bells."
2. Bob: "I saw Mommy kissing Santa Claus."
3. Alice: "Rudolph the red-nosed reindeer."
4. Bob: "I can’t let you win next turn, so... Feliz Navidad."
5. Alice: "Grinch."
6. Bob: "Oh, drat, that was clever. I would have said Joy To The World next to stop you, but I can't. You've also already taken what I would need to win."
7. Alice: "If you can't go, then I win by default!"

The game was now over. They then began the second game.

1. Alice: "I'll start out differently this time. Hark the herald angels sing."
2. Bob: "In the bleak midwinter."
3. Alice: "Coventry carol."
4. Bob: "Oh, no you don't. Silent night."
5. Alice: "Drat, now I can't win. Whichever one I say, you'll take the other one!"

The moment she said that, I suddenly realized what the game was! After declaring my newfound knowledge, I was officially entrusted with the Christmas secret. Now you have to do the same.

What is the name of the game Alice and Bob were playing, and what are the rules? A complete answer will also explain the logic of their moves, the conditions of victory and defeat, and their comments each turn.

(Note for clarity: Punctuation and capitalization are not important, only the words themselves.)

Inspired by this wonderful puzzle by @bobble, and this classic puzzle by @Stiv.

Hint #1:

"What did you mean about a paper and pencil being helpful, Bob?" I asked curiously, as we made a fresh batch of hot cocoa afterwards.

"Well, it'll help keep track of the score, and if anybody goes over," he said. "But if you get the rules, you could just as easily keep score in your head. And if you really want to game the game, you could draw a specific thing to help you strategically know what to say."

• I knew it was going to be something with a simple game... Maybe I just have the tingling feeling its rot 13 (Gvp gnp gbr) Commented Dec 11, 2021 at 22:57
• Mr. Sensitive and his tingling feelings. Commented Dec 12, 2021 at 13:07

Edit 2:
The game they're playing is

Tic-tac-toe with magic squares (not sure if there's an official name for it)

The rules are

The grid is a 3x3 magic square as shown
2 7 6
9 5 1
4 3 8

The number of syllables in each Christmas song name determines which position they want to play in

A player wins if their total of 3+ numbers equals 15

A player cannot make a move to make their total exceed 15

Let's analyze the first game. Alice starts with "Jingle bells."

This contains 3 syllables
Score: Alice 3, Bob 0

Bob goes for "I saw Mommy kissing Santa Claus."

This contains 9 syllables. Alice can win by either going 4+8 or 5+7. This didn't seem like an optimal move by Bob since he left both options open
Score: Alice 3, Bob 9

Alice responds with "Rudolph the red-nosed reindeer."

Contains 7 syllables. Alice goes for the option 5+7 for winning
Score: Alice 10, Bob 9

Bob's turn next and he says "I can’t let you win next turn, so... Feliz Navidad."

We can see that Alice would win if she played 5 next turn so in order to prevent that, Bob went for "Feliz Navidad" which contains 5 syllables
Score: Alice 10, Bob 14

Alice responds with "Grinch."

This contains 1 syllable
Score: Alice 11, Bob 14

Bob says "Oh, drat, that was clever. I would have said Joy To The World next to stop you, but I can't. You've also already taken what I would need to win."

At this point Alice needs 4 to win and Bob suggested playing a song with 4 syllables but he is already at 14 and he cannot exceed a score of 15. He also mentions that Alice has already taken his chance of winning by playing the 1 since 15-14 = 1. That's why it was a clever move

Alice finally says "If you can't go, then I win by default!"

Since Bob's only move is 1 and it is already taken. Alice declared her victory over Bob in game 1 since he cannot play a valid move

Let's go over game 2. Alice starts with "I'll start out differently this time. Hark the herald angels sing."

This song name contains 7 syllables, which indeed is different from her 3 start the previous game
Score: Alice 7, Bob 0

Bob responds with "In the bleak midwinter."

This song name contains 6 syllables. Alice's winning positions are 2+6 or 3+5. In this case Bob chooses to shut off 2+6 by taking the 6
Score: Alice 7, Bob 6

Alice goes with "Coventry carol."

It contains 5 syllables. Since 2+6 is shut off by Bob, her only choice is to try for 3+5
Score: Alice 12, Bob 6

Bob responds with "Oh, no you don't. Silent night."

We can see that Alice wins if she played 3 in the next round, which explains Bob's comment "Oh, no you don't". In order to prevent this, he plays a Christmas song which contains 3 syllables
Score: Alice 12, Bob 9

Alice finally says "Drat, now I can't win. Whichever one I say, you'll take the other one!"

At this point Alice's only winning position is to play 1 and 2 consecutively but she realizes that Bob will just play the other one and seal off her only winning chance. Thus she declares that she can no longer win in this game

• You are almost completely correct! While the game is essentially a form of Tic-Tac-Toe, it's a more specific form of Tic-Tac-Toe that uses 3x3 magic squares, deals with sums, and has a special name - you might even call it a form of "picking"... And there is indeed a maximum number, hence Bob's remark in hint #1 about going over. If you declare the specific game I'll checkmark. Commented Dec 12, 2021 at 15:49
• You've got it now - the official name of the game is Number Scrabble/Pick15! en.wikipedia.org/wiki/Number_Scrabble Commented Dec 12, 2021 at 16:39
• @Sciborg: note that Number Scrabble allows for draws (as does tic-tac-toe), but the conversations in the puzzle seem to indicate that draws are not possible ... so perhaps Alice and Bob are playing a variant. Commented Dec 12, 2021 at 18:52
• @GregMartin Very true - I tried to avoid having the example games end in a draw in favor of leaving strong indicators about why X person can now no longer win and so on. Commented Dec 12, 2021 at 19:29
• @Sciborg: FYI I did a brute-force calculation of all 1910 legal game states of this variant, and it's a second-player win. I believe a valid winning strategy for Bob is "block if necessary, otherwise choose the smallest remaining number"—apparently this holding pattern makes Alice run out of moves before Bob. Commented Dec 12, 2021 at 20:58