Over the Twelve Days of Christmas, My True Love Sent to Me 364 Presents

Puzzle 1:
On day 12, my true love sent to me 78 presents (1+2+3+4+5+6+7+8+9+10+11+12).

Day 1: 1 Day 2: 3 Day 3: 6 Day 4: 10
Day 5: 15 Day 6: 21 Day 7: 28 Day 8: 36
Day 9: 45 Day 10: 55 Day 11: 66 Day 12: 78
These are known as triangular numbers. Adding up these twelve numbers gives the total:

Over the twelve days of Christmas, my true love sent to me 364 presents.

Puzzle 2:

Puzzle 3: There's a few ways to complete the Christmas trees so each line has the same total. Here are just two:

Puzzle 4:
Question
12 carrots = enough for noses for 12 snowmen
25 sticks = enough for arms for 12 snowmen (and one left over)
45 raisins = enough for mouths for 11 snowmen (and one left over)
65 buttons = enough for eyes and coats for 13 snowmen

So Ben can build 11 snowmen. He'll run out of raisins if he tries to make any more than that.

Challenge
31 carrots = enough for noses for 31 snowmen
73 sticks = enough for arms for 36 snowmen (and one left over)
110 raisins = enough for mouths for 27 snowmen (and two left over)
165 buttons = enough for eyes and coats for 33 snowmen

So Ben and Ruby can build 27 snowmen. After that they'll run out of enough raisins to make a 28th.

Puzzle 5
There are some amazing patterns you'll spot if you play this game.
Multiples of four are important. Player Two can always win.
If Player One takes one present, Player Two takes three. If Player One takes two presents, so does Player Two. If Player One takes three presents, Player Two takes one.
You'll have to adapt your strategy if there are a different number of presents in total or if you're allowed to take a different number of presents each time. But multiples are always really important.
Merry Christmas and a Happy New Year!