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New abstract game : Sumof2

9 replies [Last post]
Mon, 10/04/2010 - 12:00
ichbin
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Joined: 09/21/2010

Game : Sumof2

Category : Abstract strategy
Age : 6 and up
Playing time : 20 minutes
Number of players : 2
Mechanic : Numbers

Components :

Board 6x6
36 pawns numbered from 1 to 36

Goal of the game : the winner is the one scoring less points after 2 rounds

Rules of the game

The game is played in 2 rounds.
The first round one of the 2 players play first
The second round is up to his opponents to play first.
The score is established after the 2 rounds.

Set up

The board is placed between the 2 players.
The board is filled randomly with all the 36 pawns faced up.

Gameplay

Pick the first to play
Players aternate picking 3 pawns form the board
Player in turn pick 3 pawns such as the value of one them is equal to the sum of the values of the 2 others.
Example :
Player pick 20, 15,5. The move is legal because 20=15+5
Player pick 10,7,18. The move is illegal because no number is sum of the 2.
10+7=17 not 18.

End of the game
The game ends when a player in turn can not find 3 pawns such as one is the sum of 2.
So the sum of the value of remaining pawns will be the score of the player who could not find a solution.
Record the score and start of the second round.

Scoring

After the second round compare the points the player with less points win the game.

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Mon, 10/04/2010 - 14:16
#1
rcjames14
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Joined: 09/17/2010
I Like The Idea

There's something elegant here.

But the way that the game is scored seems to remove strategy from the equation. If the only score that matters is the one that remains when you can't make a match, then it really only matters that you aren't the last player to play. And, as such, it seems like the game would turn into a variant of Nim.

Although I'm not sure what the precise algorithm is at the moment, it nevertheless seems to me that you could mathematically reason backwards and figure out who will win based upon who plays first. The equation would probably stem from the uneven relationship of the numbers around the mean. so 18+19 = 37, not 36. But of course, the complexity comes from that fact that 18 or 19 may have already been poached by previous combinations.

So, I think it's a really good puzzle, but my intuition tells me that it is strategically complete. However, if you introduced a different scoring mechanism, it might be less determined.

If each of the pawns has a color and there is a geometric scoring system, or if each of the pawns has a value in a range from 1 to 5, and it is based upon the numbers you collect instead, it could be interesting. After all, not being able to move on the last play means that you are down 3 items.

There may also be special 'free' move pawns that let you redeem them to collect 4 pawns on a turn instead of three or some other way for the accumulation of pawns to feed back into the capturing mechanism so that players may seize certain numbers for a later advantage.

Alternatively, you could introduce a positional element to capture which makes it harder to decipher what will be left as pawns are removed. Something like Bejeweled with numbers, I could imagine. Or maybe something else. Such as, the numbers would have to all be in the same row, column or diagonal. Perhaps adjacency matters to, so the game could end up looking like Boggle. Where every is looking for mathematically sound additions in a random 5x5 square.

This might not be where you want to go with the game, but I do think that positionality is key to making it a compelling game.

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Mon, 10/04/2010 - 21:02
#2
ichbin
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Joined: 09/21/2010
Thank you

Thank you for your comments.
In fact the scoring system have to be improved.

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Tue, 10/05/2010 - 10:12
#3
ichbin
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Joined: 09/21/2010
The game as it is now is flawed

Sorry for this idea.
With the rules above it does not work at all. I have made some simulations. It lead to bad results.
So I have to completly change the rules.
No change for the board 6x6 and the 36 pawns numbered from 1 to 36.
Only the rules are going to change.

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Tue, 10/05/2010 - 11:29
#4
rcjames14
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Joined: 09/17/2010
Alternative Mechanics

Scrabble has more As and Es in the pool than it has Ks and Qs. Why not do the same with numbers?

Essentially, you are talking about a summation mechanic that is analogous to word construction. Only instead of vowels, 1s and 2s are going to be the objects that are ten times more useful than 26s and 34s. So, they need not be evenly represented.

If you look at Poison, each of the pots needs to total up to 13 or more in order to be claimed. So, in a sense, it is a summation problem somewhat similar to yours. But, instead of evenly distributing everything between 1 and 12, Knizia has a power law distribution, with many numbers missing. There are more ones than twos, more threes than fives and the highest card is 7. Likewise, Condottiere doesn't have nearly as many 10s as it does 1s and there is a missing gap between 6 and 10. So, you might think about an inverse weighted distribution for the numbers. Especially, if you want to create a scoring system where it is more likely than not to allow all the pieces to be claimed.

And, as far as a positional aspect plays into the strategy, what if all the numbers were randomly distributed on a hex map board ( 4/5/6/7/6/5/4 ) at the beginning and you could only do summations in one of the vertical or diagonal directions? There would be a lot of tactical aspects to consider regarding the elimination of certain numbers. Of course, you'd run into the fundamental problem of reaping what you sow. Since there is no changing situation, you would be just as hurt by your actions on a subsequent turn as you would be hurt by your opponent. So, there wouldn't be much fun or excitement in removing pieces without replacement.

Bejeweled gets around this by having the matrix collapse on itself as you score, so you are always changing the location of the components. But, that would be very hard to explain and operationally tedious for a tabletop game. So, here's what I suggest instead:

Returning to the scrabble mechanic, start with a hex map that is 3/4/5/4/3 large.
With a bag full of 60 or so unevenly distributed numbers between 1 and 20, pull 19 at random to initially populate the board.
Then each player draws a hand of 5 tiles and someone goes first.
Play as follows:
You may capture any two or more tiles on the board in a row if you play an equal number of tiles from your hand which add up to the same total. So, you could replace a 6, a 7 and a 10 with a 20, a 2 and a 1. The order you do so, is up to you. You put the numbers you collect in your score pile and draw new tiles to refresh your hand.
The game continues until the bag runs out and all players pass.
You could either score simply by counting up the number of tiles you collected. Or, you could give each tile a value like in scrabble with harder to claim numbers valued higher.
The player with the highest score wins.

What do you think?

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Tue, 10/05/2010 - 14:32
#5
ichbin
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Joined: 09/21/2010
Thank you for your suggestions

but the problem is more complicated than it seems.
If I change the rules and I can still do it but the rule will be hard to aplly.
I'm trying to find something very simple.
I have to rethink all the game except the components.
Instead of a=b+c I could change it to the sum of two have to be multiple of the third one. It is not easy for people with weak background in mathematics to understand it.
The scoring system will be then more complex and so on.
The only solution is to rethink the game.
It is going to take a week or more to do that.
Anyway thanks for all your comments.

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Wed, 10/06/2010 - 09:41
#6
ichbin
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Joined: 09/21/2010
Minor change

Yesterday I have found an idea implying a minor change.
I will test it during one or two days to see if it works.
I will then publish the version 2.

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Wed, 10/06/2010 - 20:55
#7
Steve
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Joined: 07/29/2008
Possible Modification

I was thinking about this game for a while and my thoughts (in no particular order) were these:

1). Instead of a point-based system for victory, why not a "Player who plays last successfully, wins" victory?

2). Players could have "+" or "-" cards that they could play so that it just wouldn't be "sum of 2" but could also be "subtraction of two," multiply of two" and even "division of two" (the + and - could be tilted for X and /).

3). The opponent gets to choose whether the player receives a "+" or "-" card. The player can then choose if they want it to be a + or X, or a - or /. If a player succeeds in creating a valid equation (ex. 5 + 7 = 12), the card is discarded. The player then gets to decide which of the three numbers to put back onto the board. Then it's the opponent's turn.

4). Maybe introduce a mechanism for more complex equations, like 20 * 6 / 10 = 12 so that it's not just 3 numbers removed per turn.

5). Maybe add modifiers like, after your opponent chooses their numbers, you can force one of the numbers to be a different base, like base-2 (ex. 11 in Base-2 is 3 in Base-10) so that numbers with less probable use can be evened out.

Anyway, good luck with your game.

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Wed, 10/06/2010 - 22:04
#8
ichbin
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Joined: 09/21/2010
Thank you for all your suggestions

Steve wrote:
I was thinking about this game for a while and my thoughts (in no particular order) were these:

1). Instead of a point-based system for victory, why not a "Player who plays last successfully, wins" victory?

2). Players could have "+" or "-" cards that they could play so that it just wouldn't be "sum of 2" but could also be "subtraction of two," multiply of two" and even "division of two" (the + and - could be tilted for X and /).

3). The opponent gets to choose whether the player receives a "+" or "-" card. The player can then choose if they want it to be a + or X, or a - or /. If a player succeeds in creating a valid equation (ex. 5 + 7 = 12), the card is discarded. The player then gets to decide which of the three numbers to put back onto the board. Then it's the opponent's turn.

4). Maybe introduce a mechanism for more complex equations, like 20 * 6 / 10 = 12 so that it's not just 3 numbers removed per turn.

5). Maybe add modifiers like, after your opponent chooses their numbers, you can force one of the numbers to be a different base, like base-2 (ex. 11 in Base-2 is 3 in Base-10) so that numbers with less probable use can be evened out.

Anyway, good luck with your game.

Thank you for all your suggestions and your interest for my game.
We have to keep in that lot of people are not familiar with mathematics as you or as me.
I changed the rules of Sumof2.
I still need 1 ou 2 days of simulation.
Soon I will publish the version2.
Anyone 8 years old and up could play the game.
My mistake was that I did not simulate for people with high level of strategy. Rules were simple but if you want to reach a minimum of points you have to play almost randomly.
If any of the players play smartly then almost 15 out of 36 pawns will remain on the board. I expected 3 to 6 pawns.
Now with the new rules it will be easy for any player to put some light strategy in place.
Sum of 2 will remain but instead of removing 3 pawns a player will remove only one and switch 2.
The board will be divided on 2 camps.
The choice of the 3 pawns will be little bit streched.
The duration game will double.
I do not want to add lot of rules my goal is to keep the game with 3 or 4 rules.
Later I will add variants for experts.

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Wed, 10/06/2010 - 22:54
#9
rcjames14
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Variants

Steve's suggestion sounds like it might make for a good educational game. Giving kids cards to play and allowing them to score points if they are able to manipulate numbers on the board with mathematical functions would be an excellent learning tool. Kids would then be forced to understand the relationship between numbers in order to collect more tokens/pawns/points.

Beginners could start with addition or subtraction functions and additional functions could be added to the deck as kids learn them. Crafty kids might be able to use more complicated functions to create longer strings of math to score more points and the competition for claiming tokens may just be entertaining enough to overlook the educational aspect.

I think in this case, you'd want to have a small pool of maybe 7 numbers and a hand with 6 cards/functions. After you play a function and collect the numbers you used, you refresh both the pool and your hand. So, there's always 7 numbers and 6 cards in hand. Keep playing until all the tokens or cards are gone. Then total up who's collected the most to see who won.

Great idea!

As you know I have my own idea for how you might merge this design with the scrabble mechanic. So, there are just so many directions to go with this.

I look forward to seeing ichbin's new and improved version. And, I'll try to create a rules sheet with diagrams for the version I have in mind by tomorrow to counterpost.

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