Michael Birch
@birchmd.bsky.social
Software developer; math enthusiast; he/him; MTG casual;
https://github.com/birchmd
https://www.typedriven.ca/portfolios/
https://github.com/birchmd
https://www.typedriven.ca/portfolios/
I know power cube is all the rage for limited players on #MTGArena but I was in the mood for a relaxing quick draft and I managed to get 7 wins!
November 9, 2025 at 11:20 PM
I know power cube is all the rage for limited players on #MTGArena but I was in the mood for a relaxing quick draft and I managed to get 7 wins!
I guess this hand would be spookier if it was all Swamps, but 7 lands is still pretty scary! Happy Halloween!
October 31, 2025 at 3:45 PM
I guess this hand would be spookier if it was all Swamps, but 7 lands is still pretty scary! Happy Halloween!
That said, I think this simulation should still give a bit of a sense of how quickly the deck deals damage on average and how important certain decisions are. The main result is the plot showing the distribution of turn number on which the deck wins (deals lethal damage with Claim).
October 28, 2025 at 2:21 AM
That said, I think this simulation should still give a bit of a sense of how quickly the deck deals damage on average and how important certain decisions are. The main result is the plot showing the distribution of turn number on which the deck wins (deals lethal damage with Claim).
#MTGArenaStats during Edge of Eternities
October 26, 2025 at 1:33 PM
#MTGArenaStats during Edge of Eternities
It's no secret now that blue-red spells is a good deck in #MTGxFINALFANTASY draft. But it's still satisfying when a good version comes together and you get 7 wins on #MTGArena !
July 15, 2025 at 12:48 AM
It's no secret now that blue-red spells is a good deck in #MTGxFINALFANTASY draft. But it's still satisfying when a good version comes together and you get 7 wins on #MTGArena !
Ah Chocobo Spells, a classic archetype of #MTGxFINALFANTASY draft. Still good enough for 7 wins on #MTGArena
All joking aside, this deck was super fun to draft and play. And FF is a fun draft format because it has synergies allowing you build decks outside the seeded archetypes.
All joking aside, this deck was super fun to draft and play. And FF is a fun draft format because it has synergies allowing you build decks outside the seeded archetypes.
June 28, 2025 at 2:42 AM
Ah Chocobo Spells, a classic archetype of #MTGxFINALFANTASY draft. Still good enough for 7 wins on #MTGArena
All joking aside, this deck was super fun to draft and play. And FF is a fun draft format because it has synergies allowing you build decks outside the seeded archetypes.
All joking aside, this deck was super fun to draft and play. And FF is a fun draft format because it has synergies allowing you build decks outside the seeded archetypes.
My first 7 wins in #MTGxFINALFANTASY draft on #MTGArena !
June 20, 2025 at 2:25 AM
My first 7 wins in #MTGxFINALFANTASY draft on #MTGArena !
The results are shown in the figures below. The alt text of the images discusses the methodology in greater detail.
![This figure shows the result of simulating 20 million games of War. To describe what is being plotted we first need to know the mathematical model I used for the analysis.
Let's assign each card in the deck a value 2 through 14. Then the outcome of a round of the game is determined solely by comparing these values. And we can measure how well a player is doing by considering the total value of cards in their deck. A value of 416 means that player has all the cards. We also track the the number of cards each player has. A player with all the cards as 52.
Without loss of generality, we can focus on player 1 since if player 1 has a deck value V containing N cards then player 2 has a deck value of 416 - V and 52 - N cards, so dynamics of player 2 are easily deduced from the dynamics of player 1.
To make our lives easier, we rescale the variables: x = V/416 and y = N/52. Note x and y are both in the range [0, 1].
We'll derive equations for how x and y change in the next figure, but for now we have enough context to say what is plotted in this figure. It is the (x, y) state space for the games. Each colored line is the average over many games with the same (V, N) initial conditions. All lines start at y=0.5 since each player begins with half the deck. They can start at different points in the x direction because the cards are randomly assigned to the players (so player 1 may get more or less high-value cards).
All the trajectories move diagonally until they approach the line y = x, then they all follow this line to the center point (0.5,0.5). Thus we see the games evolving to this steady state where they will stay forever, as described in the main post.
Now let's see if we can explain this numerical result with our mathematical model.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreiawvugakbtqexxvw3fribja466mxhs6xscqwz7onsgs2rejhziuva@jpeg)
![To explain this plot we develop our mathematical model further. Recall from the previous figure we have the state variables V and N, and their rescaled versions x and y.
Suppose player 1 has a probability p of winning the next round of the game. Clearly p depends on V and N, we'll come back to that later (that's what this plot is about). For now we have the following equation for how V changes:
dV = p (416 - V)/(52 - N) - (1 - p) V/N
i.e. if player 1 wins the round then they add value to their deck equal to the average value of their opponent's deck, and if they lose then they lose value from their deck equal to their average value.
Similarly, the equation for how N changes is:
dN = p - (1 - p) = 2p - 1
i.e. N goes up by 1 with probability p and down by 1 with probability 1-p.
The equations are cleaner in terms of our rescaled variables:
52dx = p[(1-x)/(1-y) + x/y] - x/y
52dy = 2p - 1.
First we can notice that there is an equilibrium (dx=dy=0) when p=x=y=1/2. I.e. both players have half the deck. And this is exactly what we observed in the numerical simulation.
The differential equations are non-linear, but we can check if this equilibrium is stable by linearizing the equations. A little calculus reveals:
26dx = (a - 1)(x - 0.5) + (b + 1)(y - 0.5)
26dy = a(x - 0.5) + b(y - 0.5)
where a = dp/dx and b = dp/dy, both evaluated at x=1/2,y=1/2.
The eigenvalues of the matrix for this linear system are -1 and a+b, so to make progress we need to know the value of these derivatives. Maybe there is a way to get them analytically, but we can get them numerically from the same data that gave the first figure.
For each (V,N) state we count how often player 1 won the round to estimate the probability p. Plotted in this figure is p as a function of x for different (fixed) values of y. Lines for values of y lower than 0.5 are redder and those higher are greener. There are multiple nearly parallel diagonal lines with positive slope.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreiho4qldfauvukhgbbnl724xzqebolu3myzuaredgpaiti7ycpenta@jpeg)
April 13, 2025 at 9:36 PM
The results are shown in the figures below. The alt text of the images discusses the methodology in greater detail.
7 wins with a blue/white artifacts deck in #MTGAetherdrift !
March 22, 2025 at 3:17 PM
7 wins with a blue/white artifacts deck in #MTGAetherdrift !
And what better way to celebrate than with some beers from Collective Arts?
I enjoyed both of these NEIPAs, though personally I liked Rare Specimen more than Deceptive Creature because the latter seemed more like fizzy juice than beer to me.
I enjoyed both of these NEIPAs, though personally I liked Rare Specimen more than Deceptive Creature because the latter seemed more like fizzy juice than beer to me.
March 14, 2025 at 11:25 PM
And what better way to celebrate than with some beers from Collective Arts?
I enjoyed both of these NEIPAs, though personally I liked Rare Specimen more than Deceptive Creature because the latter seemed more like fizzy juice than beer to me.
I enjoyed both of these NEIPAs, though personally I liked Rare Specimen more than Deceptive Creature because the latter seemed more like fizzy juice than beer to me.
For example we have the following non-trivial solution mod 11:
1^2 + 9^2 = 4^2
1^2 + 6^2 = 9^2
This solution gives the magic square of squares* shown in the image.
*Allowing Gaussian integers as squares and taking the sums mod 11.
10/12
1^2 + 9^2 = 4^2
1^2 + 6^2 = 9^2
This solution gives the magic square of squares* shown in the image.
*Allowing Gaussian integers as squares and taking the sums mod 11.
10/12
March 6, 2025 at 9:36 PM
For example we have the following non-trivial solution mod 11:
1^2 + 9^2 = 4^2
1^2 + 6^2 = 9^2
This solution gives the magic square of squares* shown in the image.
*Allowing Gaussian integers as squares and taking the sums mod 11.
10/12
1^2 + 9^2 = 4^2
1^2 + 6^2 = 9^2
This solution gives the magic square of squares* shown in the image.
*Allowing Gaussian integers as squares and taking the sums mod 11.
10/12
I started with some linear algebra to constrain the 3x3 grid (ostensibly 9 variables) down to just three variables. Then I noticed that I had re-derived the same formula used by Lucas[3] (of Lucas numbers fame). This formula is very pretty in how symmetric it is and gave me an idea.
2/12
2/12
March 6, 2025 at 9:36 PM
I started with some linear algebra to constrain the 3x3 grid (ostensibly 9 variables) down to just three variables. Then I noticed that I had re-derived the same formula used by Lucas[3] (of Lucas numbers fame). This formula is very pretty in how symmetric it is and gave me an idea.
2/12
2/12
I went 0-4 in Day 2 of the #MTGArena Open. I don't think my deck was incredible (lacking on removal for example), but it would have been nice to win at least one game. Oh well, better luck next time!
March 2, 2025 at 3:37 PM
I went 0-4 in Day 2 of the #MTGArena Open. I don't think my deck was incredible (lacking on removal for example), but it would have been nice to win at least one game. Oh well, better luck next time!
I qualified for Day 2 of the #MTGAetherdrift Arena Open! I didn't think my sealed pool was that strong, but the RB speed deck managed to go the distance!
March 1, 2025 at 9:18 PM
I qualified for Day 2 of the #MTGAetherdrift Arena Open! I didn't think my sealed pool was that strong, but the RB speed deck managed to go the distance!
#MTGArenaStats during Foundations. Looks like I played fewer games than I normally do, but hitting mythic (a while ago now) was cool!
February 20, 2025 at 9:09 AM
#MTGArenaStats during Foundations. Looks like I played fewer games than I normally do, but hitting mythic (a while ago now) was cool!
Finishing up beermas 2024 *strong*! You really feel the 10% ABV on this one.
#BeerSky
Looking back, this was a good beermas. I really enjoyed most of the beers from Brock St Brewing's Advent Calendar. I recommend giving them a try if you haven't before!
#BeerSky
Looking back, this was a good beermas. I really enjoyed most of the beers from Brock St Brewing's Advent Calendar. I recommend giving them a try if you haven't before!
December 24, 2024 at 10:46 PM
Finishing up beermas 2024 *strong*! You really feel the 10% ABV on this one.
#BeerSky
Looking back, this was a good beermas. I really enjoyed most of the beers from Brock St Brewing's Advent Calendar. I recommend giving them a try if you haven't before!
#BeerSky
Looking back, this was a good beermas. I really enjoyed most of the beers from Brock St Brewing's Advent Calendar. I recommend giving them a try if you haven't before!
How festive!
I'm generally skeptical of flavoured dark beers like this, but this one is quite nice! The chocolate comes through nicely with just a hint of the orange, though at its core this beer is just a good stout.
#BeerSky
I'm generally skeptical of flavoured dark beers like this, but this one is quite nice! The chocolate comes through nicely with just a hint of the orange, though at its core this beer is just a good stout.
#BeerSky
December 23, 2024 at 10:16 PM
How festive!
I'm generally skeptical of flavoured dark beers like this, but this one is quite nice! The chocolate comes through nicely with just a hint of the orange, though at its core this beer is just a good stout.
#BeerSky
I'm generally skeptical of flavoured dark beers like this, but this one is quite nice! The chocolate comes through nicely with just a hint of the orange, though at its core this beer is just a good stout.
#BeerSky
December 20, 2024 at 12:18 AM
I just completed "Restroom Redoubt" - Day 14 - Advent of Code 2024 #AdventOfCode adventofcode.com/2024/day/14
A very festive problem today! The ASCII art Christmas tree actually looks really good!
A very festive problem today! The ASCII art Christmas tree actually looks really good!
December 14, 2024 at 3:21 PM
I just completed "Restroom Redoubt" - Day 14 - Advent of Code 2024 #AdventOfCode adventofcode.com/2024/day/14
A very festive problem today! The ASCII art Christmas tree actually looks really good!
A very festive problem today! The ASCII art Christmas tree actually looks really good!
For day 9 of beermas, Brock St brewing has included a uniquely Canadian beer! I like the subtle maple flavour the real sap used in the brewing process imparts on the final product.
#BeerSky
#BeerSky
December 10, 2024 at 12:23 AM
For day 9 of beermas, Brock St brewing has included a uniquely Canadian beer! I like the subtle maple flavour the real sap used in the brewing process imparts on the final product.
#BeerSky
#BeerSky
On this, the fourth day of beermas, Brock St brewing has given me an amber beer! I quite enjoy ambers and this is a pretty nice one 😋
#BeerSky
#BeerSky
December 4, 2024 at 11:23 PM
On this, the fourth day of beermas, Brock St brewing has given me an amber beer! I quite enjoy ambers and this is a pretty nice one 😋
#BeerSky
#BeerSky
It's December 1st! You know what that means. It's the first day of beermas!
December 1, 2024 at 6:34 PM
It's December 1st! You know what that means. It's the first day of beermas!
First snow flurries of the season!
November 29, 2024 at 7:47 PM
First snow flurries of the season!
November 25, 2024 at 7:40 PM