I'd go with programming each one to be ID'd by their pulse signature so that the game board sends out a call to all 13 devices, and all 13 devices report back. Each device can have a unique signature in which when the signature is a match it reports back so that not all 13 are trying to report back at the same time.
So you could have a board with a matrix of coils which are used to transmit and receive the pulses to poll devices by location on X,Y and know of their placement. Without knowing your budget and skill level though I am unsure if you can achieve this or not.
But if the game board sent out a poll such as
10001 to unlock an output from device 1
10010 to unlock an output from device 2
10011 to unlock an output from device 3
10100 to unlock an output from device 4
10101 to unlock an output from device 5
10110 to unlock an output from device 6
10111 to unlock an output from device 7
11000 to unlock an output from device 8
11001 to unlock an output from device 9
11010 to unlock an output from device 10
11011 to unlock an output from device 11
11100 to unlock an output from device 12
11101 to unlock an output from device 13
And the game board waits for response and confirms receiving response and processes this data to your program or does not detect the response and so it calls out again to it until it gets the response and likely alerts to an error condition where a game piece is not responding. *If it reports back, then call to device 2 and 3 and so on and register on a matrix playing field as to where each device is.
The first bit is a wakeup and the following 4 bits are the address of the device for its unique ID to be called. To make it simple, lets say it responds with its Unique ID back after a short delay so the game board says Device 1, and 1/4 sec delay and Device 1 HERE basically at strongest reading at X=32, Y=24 as this is the coil that received the message that it is there so this is its location.
It processes this polling until all 13 devices report back. To make this work however and low cost, I would interface this with a PC, however this could also be done with a Raspberry PI I suppose. My choice would be with a PC to run the program and test it as for C++ is my programming and interface language of choice and I havent worked with Raspberry PI micro computer yet.
Not to scare you off on this, but your probably looking at an initial build cost far beyond what you have to play with in your budget. I have many many circuits drawn up that usually dont come to prototype build because I dont have the budget to make it. So it remains on paper as soemthing that would work and likely work out really well if I had funding to make it happen.
The board itself will be pretty easy to make with a group of coils mounted under the board under a thin enough material so that the magnetic transmission can pass through without too much loss. The hardest and most expensive part will be the circuits that are embedded in each of the game pieces that need to listen and respond with their name when their matching Device ID is called.
Below are methods that are not a match to your description but position detection methods.
Its too bad you dont have just 2 game pieces as for you could set one piece to South pole facing the game board and the other with North pole of magnet facing the game board and detect by pole ID.
The count of 13 in a board game I find interesting because most board games have an equal number of pieces between 2 players, so I am curious as to what type of game this is, if that info can be shared or not.
An extremely low low cost board game to keep track of pieces, now this is the absolute lowest tech board game and primitive, would be to have pieces that physically plugged into places on the game board or made electrical contact with contacts on the game board. Each of the 13 pieces can have a unique circuit and where they are placed on the X and Y is identified by their circuit signature. This could be performed for less than $100 given its so primitive and low tech. You still would want to interface this with a computer probably though to track game piece locations etc unless you can squeeze the logic into a low cost micro computer. If you already have a computer you can write a program that polls the game board for every square of the X,Y playing field and each square that has a game piece in it will report a different circuit config back to a game piece identifier/translator that is then reported back to the PC. You would need to program up the code for free on your own time to keep costs to bare minimum so that only the hardware that you need composes the costs of the build with time to build it not measured in monetary value.
The game pieces by the way to clarify how low tech they are are just bread board and traces with no components on them. Each one when in electrical contact with the game board will report back with a solid binary signature that is hard wired so if you have 5 contacts on each game piece you have your common and 4 other contacts which are the binary signature, so if contact 1 is common and its connected directly to contact 4 and 5, with 5 being the first binary digit 0 or 1 and contact 4 being the second binary digit of 0 or 2 and contact 3 being the third binary digit of 0 or 4 and contact 2 being the fourth binary digit of 0 or 8, ( contact 1 common tied directly with contact 4 and 5 reports back as hard wired Device ID # 3 )
