The initial circle of seated students is provided, which leaves Melissa as the first It. The students sharing her initial are Molly and Matilda, but only starting with Molly satisfies the other conditions. Therefore the starting positions are as follows:
As her first move, Melissa taps 4 students on the head, ending with Daisy (though she only counts every other student for her total.)
Daisy has been It once, so she taps Tanya as Buck and Lucky as Moose.
Lucky taps the next student, Matilda, to be Moose.
Matilda taps Donald as Moose. Donald’s blue cap is first passed to Melissa and ends up with Angus, the nearest boy to Melissa. The game (keeping track of student counts in parentheses) now looks like:
For the next two steps, Donald taps one of the brothers to be Moose, who becomes It and picks another brother sitting 3 students away. The only valid combination of moves is that Donald taps Huey, who then taps Dewey.
Since Huey tapped 3 students, Dewey taps 4 - ending with Daisy, who becomes the next It.
Daisy has now been It twice so she taps 3 students in total, with Daphne becoming the next Moose. During this round, Tanya and Angus exchange head accessories.
Of the 3 brothers, Daphne is closest to Louie, so she taps him to be Moose and takes his spot.
Daisy and Daphne swap head accessories. Louie one-ups Dewey by tapping 5 total students, ending with Angus. Angus then taps 3 Bucks before selecting the next boy. If Lucky is a boy, then Lucky becomes the Moose - but if Lucky is a girl, then Donald becomes the Moose.
Looking ahead a couple of steps, whoever becomes It first taps Molly, and then a student that he’d previously outrun. Lucky has only outrun Matilda, but Matilda cannot tap herself in the following round. Therefore, Lucky is a girl and Angus must have tapped Donald. Donald tries to tap Molly, but is tagged back and starts a new round by tapping 1 student, Huey, to be Moose.
The new circle looks like:
Huey taps a total of 6 students ending with Matilda, who then taps Tanya as Moose and takes her seat. (Note that Matilda drops the tens digit in her count, resetting it to 0.)
Tanya taps 5 Bucks before selecting the next girl - Molly, the 8th student away - as Moose.
Molly sees Daisy, Daphne, and Angus in a row, and picks Angus to become the next Moose.
Angus taps 3 Bucks followed by Louie as Moose and sits back in his starting location. Donald, sitting next to Tanya, tries to get his cap, but she passes it to Gus on her other side.
Louie taps a total of 7 students, ending with Daphne as Moose.
Daphne then walks a full circle before tapping Gus (14 students away.) Since she has a bad memory, she discards her previous count.
Gus then taps Dewey as Moose and takes his spot, allowing the head accessories to be evenly spaced.
Dewey one-ups Louie by tapping a total of 8 students, ending with Donald.
Donald returns to a traditional “Buck, Buck, Moose”, tapping Matilda to be Moose.
Matilda then taps and outruns Melissa, the only other student with 7 letters. During this round, Matilda again reaches double digits and drops the tens place, ending with a total count of 1.
Melissa passes Gus and Daisy wearing head accessories, then taps Louie to be Moose (again only counting every other student.) Louie taps 9 total students, ending with Lucky.
Lucky then taps Huey, who taps Gus (the 10th student away) to be Moose.
At this point, the numbers drawn at the start of the game need to be examined. Lucky is the only student who has always tapped the first student as Moose, so she drew #1. This makes Gus #7, so he taps Tanya as Moose and takes her spot.
Tanya taps 5 Bucks, then chooses Lucky to be Moose.
Lucky taps Daisy as Moose. While Daisy chases Lucky, Melissa moves clockwise into the empty spot, so Lucky needs to run more than a full circle to reach safety.
Even though Lucky was the previous It, Melissa is in the spot where the most recent Moose (Daisy) was sitting. Therefore Daisy begins the next round with Lucky, tapping 3 Bucks before tapping Gus to be Moose.
Gus storms off, and we are left with the final configuration of students and counts:
Clockwise from Donald, the reported counts (keeping in mind that Molly only reports the tens place) are:
7 - 1 - 12 - 12 - 9 - 14 - 1 - 3 - 5 - 15 - 21 - 19
Substituting these for letters gives us the final answer: GALLINACEOUS