Death Boards and Living Boards

A two-part, foolproof method for conquering rule substituions

Author

Mika MacDonald
April 16, 2023

I am revising this, bear with me y’all. I’m calling it death boarding now, because that sounds so much cooler, right? One of my students called it that, and I was hooked. Anyways, I’m making stylistic revisions too, and adding a third example. Hopefully done by weekend!

Death Boards

Hi! This is Mika. This two-part methods has led to massive accuracy and speed gains for my students, so I decided to write it up!

Let’s think about a rule substitution, and what it requires.

  • The new rule must forbid all previously forbidden boards.

  • The new rule must allow all previously allowed boards.

Thus, the key to tackling a rule substitution question is developing a method for determining whether a rule is a perfect substitute.

Let start with the first requirement. [add more here]

Therefore, the goal is to design a death board (a board not allowed, under any circumstances, under the original rules). We then determine confirm whether the substitute rule legalizes the death board.

We’re going to start by looking at 76-3-4.

Q23 involves a rule substitution. We’re replacing the rule “ “[i]f Fall contains M,, Spring contains N” (paraphrasing for legal reasons).

I would translate that as:

MF —> NS 

In this game, there are only two groups (Fall & Spring), and we are looking at only two pieces (for now). As a result, there are 4 placement scenarios. Let’s break this down, visually.

the living boards
and the death board

The current rule allows three of these scenarios, and forbids the final one:
MF + NF.

This is our death board. If a replacement rule allows MF + NF , it is not a perfect substitute and must be discarded.

Death-Boarding 101

First, we need to step back and examine the rest of the rules.

  • Pieces (6): K L M N O P

  • Groups (2): F | S

  • Exact Use (Pieces): x1

  • Minimum Use (Groups): undefined by structural rules, but automatic minimum of 1 piece by virtue of game rules.

Now let’s take a look at the other game rules!

We’ve identified our death-board so now we need to apply the remaining rules and construct our complete death-board for this question.

original death board

  1. M is in Fall, so P must be in Spring.

  2. N is in fall and K must be with it, so K is in Fall.

  3. K is in Fall, which forces O in Fall.

  4. L is unassigned.

completed death board

Because L is a floater, I want to indicate its flexibility. By hand, I would simply write the L on top of the middle line of the diagram. Now write this diagram five four times, then turn back to our rule substitution answers.

  1. LF —> MS

If we put the L in Spring, all rules are being followed, but M and N are still together in the fall.

Original Rule: Forbidden
Substitute Rule: Allowed

Thus, (A) is wrong. The new rule permits the forbidden combination and, therefore, is not a perfect substitute for the original rule.

  1. NF —> PF

Now this is interesting! M ≠ P still applies. So if NF —> PF, then NF —> MS

This forbidden scenario is now forbidden. (B) provides a rule that categorically prevents M & N from being together in Fall. Hurrah!

Original Rule: Forbidden
Substitute Rule: Forbidden

DO NOT STOP HERE (if you have the time). On advanced rule substitution questions, two or three answer choices might survive the Death Board test. That’s where the Living Boards comes into play (we’ll get there really soon).

The best strategy is to immediately move on to the next answer choice and continue testing the Forbidden Combo. If no other answer choice survives this test, then we’re done! If more than one choice survives, then we loop back to apply the Permissive Combo test to each remaining answer choice.

  1. MS —> PF

M&P are already required to be opposite, so this rule does literally nothing.

  1. NS —> MS

Doesn’t prevent MF + NF

Original: Forbidden
Substitute: Allowed

The rule is only triggered if N is in spring; thus, it does nothing to prevent the forbidden combination. (D) is wrong.

  1. OS —> NS

Doesn’t prevent MF + NF

Original: Forbidden
Substitute: Allowed

O is in the Fall, so the rule isn’t even triggered!

If you noticed, I didn’t even have to construct multiple boards. The exact same board was sufficient! That’s the point: while this test is non-intuitive initially (double falsification is a concept that most humans never use), it’s simple and time efficient to use.

This, however, is not the most complex rule substitution question; we didn’t even need to use our Living Boards.

Living Boards

This test is the flipside of the Forbidden Combo test. In this test, we ask whether the new rule forbids a board that was previously legal.

Let’s turn to 81-4-4. Only 29% of students get question 23 correct. Crazy! But I promise we can make it much easier.

[explain more of the fundamentals here]

When you try this yourself, play through the entire game. The boards you save as you answer other questions are our living boards! You need boards to do this. I recommend you circle every correct board you create, and X out all broken boards you created.

Let’s check our structural rules:

  • Pieces (4): F G H I

  • Groups (3): M | S | T

  • Minimum Use (Pieces): x1

  • Exact Use (Groups): 2 pieces/group

  • Distribution of pieces: 2-2-1-1 or 3-1-1-1 (without looking at game rules)

Now our game rules:

Inferences

Ix2 means that 3-1-1-1 is impossible. Piece distribution is always 2-2-1-1

Ibanez always visits S. see Q19.

Revised Pieces

F G H I1 I2 ?

Again, we start by identifying a forbidden scenario for testing. The rule we’re replacing is F ≠ H.

In a multigroup game, it can hard to identify the optimal forbidden board. We’re going to put F&H together in M. We’re doing this because we don’t want to put Glesson in Manilla, because that would force H to be used twice (and thereby constrain the testing board).

Once F&H are placed, the two I pieces must be placed in S & T. Gleeson cannot visit S, but must visit at least one city, so Gleeson must be in T. We have no rule forcing S’s second spot. Thus, F & H float together in that spot.

the forbidden base board

(A) Permissive Combo Knockout

G ≠ I 

This blocks F&H completely! At least one spot in each city is taken and, therefore, F can never be with H. Thus, this answer choice survives the Forbidden Combo test.

However, I generated this living board when answering question 18:

Original Rule: Allowed
Substitute Rule: Forbidden

Under the original rules, this board was allowed. Ix2, F and H are apart, the Gleeson-Haley rule is fulfilled, and Gleeson is not in Sydney.

Thus, (A) fails the one of our Living Boards. G + I was legal under the original rule. The new rule prohibits something that was previously allowed and, therefore, is not a perfect substitute.

(B) Forbidden Combo Knockout

FS —> HT 

We can easily dismiss this rule by using our initial diagram. We just need to fix H as the second position in S.

Original: Forbidden
Substitute: Permitted

The substitute rule fails to trigger at all, and F + H are still together in M. (B) fails the forbidden combo test.

(C) Forbidden Combo Knockout

HT OR FT (I capitalize OR to indicate a disjunctive)

Here’s what we’re going to do. We’re going to take our base diagram and move everyone down a city.

Original: Forbidden
Substitute: Allowed

All rules are satisfied, and one of F or H is in T. F&H are still together, and, therefore, (C) fails the Forbidden Combo Test.

Visualizing swaps on your prior boards is a skill worth investing in. Think consciously about which individual pieces can swap, which formations of pieces can swap, and when groups are and are not relevant to a swap.

In this game, the only city-based rules are G≠S and GM —> HT. I could see that rotating the rows leads to GM, and that we could play IH in T, fulfilling that rule. G isn’t in S, so that rule is irrelevant. All city rules are fulfilled, and all other rules were already fulfilled in the original board.

I always write the final board on paper and visually confirm my mental calculations. I always always always do this. I have de minimis visual imagination (it’s what held me back in chess), so seeing the final product on paper is critical (for me) in avoiding errors.

(D) Permissive Combo Knockout

-F —> H

Let’s take a look back at that board I generated while answering #18. Neither F nor H are in M.

Original: Permitted
Substitute: Forbidden

This was previously an allowed outcome. Again, the new rule would forbid something that was previously allowed!

(E) The Right Answer

-I —> G

This one is off to a good start: At least one slot in each city is occupied by I or G, which prevents F + H from ever occurring.

All I’m going to do next is look at the legal boards I constructed as I played this logic game. All six are consistent with the above rule (If I isn’t in a city, G is).

I’m done here. I absolutely proved the other answers wrong, and this rule doesn’t conflict with any prior boards.

One for the Road

Let’s take a look at one more toughie (but less tough): 71-3-16

Again, I think it’s best you work through the whole game first. Having solved boards to compare against is invaluable in applying the Permitted Combo Test.

[TBC]

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