Here’s a fun game you can play with another person.

The game starts on January 1. Each of two players takes turns calling out another date. The new date has to be a later date in the year with either the same month OR the same day (from January 1, a player can call out a later day in January or another month with the day 1 like February 1, March 1, etc.).

The person who calls out December 31 wins the game.

Who has the winning strategy, the first or second player? Does it matter if the game is played in a leap year?

This game was covered by Grey Matters in 2010, who communicated it to ScamSchool who made a video.

I have posted a video that explains the winning strategy.

**Can You Solve The Race To December 31 Riddle?**

Can you figure it out? Keep reading for my explanation.

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.**Answer To First To December 31 Wins**

The winning strategy belongs to the first player, and it does not matter if it is a leap year.

The first player should call out January 20 on the first turn. The second player then names a later day in January or another month with the day 20.

Regardless, the first player can then name one of following dates (month/day), and continue to do so no matter what the second player does.

1/20 (first turn)

2/21

3/22

4/23

5/24

6/25

7/26

8/27

9/28

10/29

11/30

12/31

The pattern is:

day = month + 19

Played correctly, the first player always calls out December 31.

**Solving the game**

Understanding the strategy is easy enough. But how could one derive the strategy?

The game is an example of a well-known combinatorial game called Nim, in which two players remove objects from different piles of objects. Nim is a completely solved game for any number of objects and piles, so it is always possible to figure out a winning strategy for one of the players.

In specific games, we can use backwards induction, which means to think ahead and reason backwards. So let us start from the end of the calendar year.

Let’s start out by solving for winning dates. By rule, December 31 is a winning date.

12/31

If you name any other date in December, you would lose, as your opponent could then name 12/31. So all other dates in December are losing dates. You also want to avoid calling earlier months with the day value 31–as your opponent could then name 12/31.

What about November? The next date to analyze is 11/30. If you could call that out, there is no later date in November, so your opponent would have to name 12/30. This allows you to call 12/31 and win. So 11/30 is another winning date. If you can name it, you are sure to win.

11/30

12/31

Now we reason backwards. Similarly any earlier date in November is losing, as are the dates 10/31 and 10/30. So you want to name one day backwards, which is 10/29.

If you could name 10/29, your opponent would have to name 10/30, 10/31, 11/29, or 12/29. From 10/30 and 11/29 you can get to 11/30, and from 10/31 or 12/29 you can get to 12/31. In other words, if you can name 10/29, you can always get to a winning date.

10/29

11/30

12/31

Now we can extend the logic that the next winning date is 1 month backwards and 1 day backwards. So we get the winning date 9/28, then 8/27, and so on, until we get all of the winning dates.

1/20

2/21

3/22

4/23

5/24

6/25

7/26

8/27

9/28

10/29

11/30

12/31

The first player can name 1/20 on the first turn, and then keep naming a winning date for any choice that the second player is allowed to make.

The pattern in the dates is:

day = month + 19

**Sources**

Grey Matters

http://headinside.blogspot.com/2010/06/scam-school-meets-grey-matters.html

Scam School

https://www.youtube.com/watch?v=J3_XJtscvnQ

Go here to read the rest:

The Race To December 31 – Sunday Puzzle