This document outlines the fundamental mathematical systems that underlie gameplay in Nubby's Number Factory, independent of specific item or perk interactions.

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- The board consists of 21 peg slots.
- Pegs have values that are powers of two: 1, 2, 4, 8, 16, 32, ...
- Pegs are popped when Nubby hits them or when a summon or item causes a pop.
- When a peg is popped, it is reduced to half its value (rounded down), and the value popped is added to the score.
- A peg is removed from the board entirely once its value becomes 0. This is called a full pop.

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Popping a peg of value `v` gives the following total score (if popped fully down to 0):
This is the maximum score that can be obtained from fully popping a peg.

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A restock occurs when the player's score in a round equals or exceeds the current quota.

Each restock performs the following operations in order:

1. For each empty slot, insert a new peg of value:

2. Perform a single merge pass:
- For every value `v`, if there are at least two pegs of that value, one pair will merge into a single peg of value `2v`.
- Only one merge pass is performed. Merges do not recurse.
- Merge order is not adjacency-based, and pegs can merge even if they are in opposite positions.

3. Award the player 1 coin per restock.

Each restock increases the total peg value on the board, leading to higher quotas.

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The quota is calculated once at the beginning of each round and does not change mid-round.

Let `V_total` be the sum of all peg values on the board, computed using:

So,

Let `multiplier` be the value from the following table based on number of shops visited:


Then the target score (quota) is:

VERY IMPORTANT NOTE - THIS MEANS (as of patch 1.3), YES, GOING TO THE BLACK MARKET DOES NOT TICK UP YOUR QUOTA MULTIPLIER. YOU SHOULD BASICALLY ALWAYS BE GOING TO THE BLACK MARKET.

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If all pegs on the board are fully popped (board is empty), a Board Clear occurs, which doubles your total score.

This is a powerful end-of-round reward, and an important consideration for score efficiency.

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Each peg has the following properties:

- Value `v`, where `v` is a power of two.
- Full score potential = `2v - 1`
- When popped, adds `v` to score and changes to `v / 2`
- When `v == 1`, popping removes the peg (full pop)
- Pegs with same value may merge during a restock (once per restock)

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Each restock increases the total value on the board through:

- Adding new pegs of fixed value per round tier
- Merging existing pegs into higher values

Let `r = round number`, then new peg value on restock is:

Over multiple restocks, the board's average peg value increases geometrically, especially if merges are efficient.

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The game is divided into 4 acts:

- Act 1: Rounds 0–20 (Early Game)
- Act 2: Rounds 21–40 (Midgame)
- Act 3: Rounds 41–60 (Lategame)
- Act 4: Rounds 61–80 (Endgame)

Score expectations, peg values, and quota sizes increase significantly with each phase.

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- Peg merge behavior is deterministic unless the board has been shuffled.
- Board shuffles introduce positional randomness, but not value randomness.
- Peg pops are determined by Nubby's physics and bounce angle, which includes slight RNG.

Given the same board state and sequence of actions (with no shuffling), outcomes are deterministic.

The following table will show you the board state assuming you only ever manage a single restock, with no items or anything else. You can consider this toy model a "lower bound" and analyze it for behaviors.

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This guide provides a rigorous mathematical breakdown of peg popping mechanics and their role in score generation, item/proc activation, and viability in building toward quota-clearing runs. This section builds on the foundational mechanics described in Basic Nubbyology, focusing specifically on pop-based systems.

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Each peg begins with a value `v`, where `v` is a power of two (1, 2, 4, 8, 16, ...).

Each time a peg is popped:

• You gain score equal to its current value.
  
• The peg’s value is halved (rounded down).
  
• If its value reaches 0, the peg is removed from the board ("full pop").

The total score from a full pop of a peg is the sum of its value over successive halvings:


This formula gives the upper bound on the score that can be gained from a single peg, assuming it is popped all the way to 0.

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To estimate the total board score potential:

• Let `v_i` be the value of each peg on the board (up to 21 pegs).
  
• For each peg, compute `2v_i - 1`.
  
• Sum these to get `TotalFullBoardScore`.

This gives the theoretical maximum score for a round, before any multipliers or bonuses are applied.

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From the quota multiplier table, we know that:


If a player fully clears the board, they receive a 2x board clear bonus:


Therefore:


This implies:

• Any build that consistently fully clears the board will, by definition, meet or exceed the quota (outside of endless mode).
  
• Even as the multiplier increases with more shop visits, the board clear bonus ensures survival is possible with perfect clearing.
  
• Builds that do not consistently clear the board must rely on supplemental score sources (items, perks, etc.) to remain viable.

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Each pop is also a resource used to:

• Trigger items (e.g., every 5 or 10 pops)
  
• Trigger perks (e.g., every 10, 15, or 20 pops)
  
• Chain summoner effects
  
• Contribute to restock chains

Thus, optimizing for total pops per round has benefits beyond score alone. For example:

The game strongly rewards recursive pop loops, even when individual pop values are low.

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Below is a non-exhaustive list of systems that rely on peg pops directly:

Pop-Triggered Items:

• Flutty: every 2 pops
  
• Cactus: every 5 pops
  
• Squirmy: every 5 pops
  
• Pants: every 10 pops
  
• Crazy Straw: every 10 pops
  
• Ton of Feathers: on full pop

Pop-Triggered Perks:

• Drainer: every 10 pops
  
• House'O'Cards: every 15 pops
  
• Eggy: every 20 pops

Each of these may increase score directly or indirectly by generating more pops, spawning summons, or force-triggering other items.

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• Popping is both the core scoring mechanism and the primary trigger system for a wide range of effects.
  
• Each full pop provides a score of 2v - 1, meaning that even low-value pegs contribute significantly over time.
  
• Consistent board clears ensure quota matching through the 2x multiplier, making them a mathematically viable long-term strategy.
  
• Non-clear builds must compensate with external score generation to remain sustainable.
  
• Pop triggers and recursion are central to score acceleration, restock chaining, and late-game viability.

This concludes the core math on peg popping. Let's get into some case studies.

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This section models the well-known "House of Cards Pants Loop", where Pants is used to recursively pop the board via repeated triggers, enabled by the House'O'Cards perk.

Pants:

• Triggers every 10 pops.
  
• Each Pants item pops 1 random peg.
  
• That peg, when popped, gives 2x score (unupgraded) or 4x score (upgraded).
  
• Each Pants trigger adds exactly one pop per Pants item.

House'O'Cards Perk:

• Triggers every 15 pops.
  
• Force-triggers all "X pegs popped →" items, including Pants.
  
• Multiple copies of the perk stack: if you have `h` copies, it will retrigger Pants `h` times every 15 pops.

Important Notes:

• Pants do not trigger individually; all Pants activate together at every 10-pop milestone.
  
• Each Pants activation results in `p` pops, where `p` is the number of Pants equipped.
  
• All of those pops count toward the next pop thresholds (10, 15, etc.).

Let:

Trigger chain:

1. Reach 10 pops → all `p` Pants activate → +`p` more pops
2. Reach 15 pops → all House'O'Cards perks activate → force-trigger Pants `h` times
3. Each forced Pants trigger → `p` additional pops
4. These forced pops count toward next 10/15 thresholds

This creates a pop loop.


To go infinite (clear the entire board without further player input), the loop must produce more pops per cycle than it consumes.

Let’s compute pops produced per cycle:

At pop 10:

At pop 15:

Total pops gained from one 15-pop cycle:


Pops consumed to reach the next cycle: 15

To go infinite:


To find the minimum number of Pants and House'O'Cards to go infinite, solve:


Try integer combinations:

Case 1: 5 Pants
Infinite at 5 Pants and 2 House'O'Cards

Case 2: 3 Pants
Infinite at 3 Pants and 4 House'O'Cards

Case 3: 7 Pants
Infinite at 7 Pants and 2 House'O'Cards


The loop becomes infinite when:


This models how many total pops are generated per loop cycle versus how many are required to trigger the loop again.

Once this condition is met:

• The board will be cleared through recursive Pants activations.
  
• The pop count will increase geometrically.
  
• The loop terminates only when the board is empty.
  
• This guarantees a Board Clear and 2x multiplier.

If you are having trouble beating the game, this is a decent strategy to fish for. Pants is a good item on its own.

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This section analyzes the behavior of Disguise Glasses, particularly when stacked or upgraded, and how they can be used to facilitate recursive board-clearing loops.


Disguise Glasses:

• Trigger condition: A peg of the current highest value is popped.
  
• Effect (base): Pops 2 random pegs.
  
• Effect (upgraded): Pops 2 random pegs, then repeats this action once more (total of 4 random pegs popped).
  
• Stacking multiple copies results in multiple triggers on a single highest-value peg pop.

• Only the highest-value peg(s) on the board will trigger Disguise Glasses.
  
• If there are multiple pegs tied for the highest value, popping any of them will also trigger the item.
  
• Random peg pops caused by the glasses may pop other pegs tied for highest, recursively triggering the effect again.
  
• Pops occur in sequence, and if a peg is reduced to tie for highest (e.g., 1024 → 512), it may then contribute to additional triggers.

Suppose the board contains:

1. You pop the 1024:
- It becomes 512
- Glasses trigger: pop 2 random pegs

2. If one of those pops is a 512:
- That 512 is now a "highest" peg
- It triggers Glasses again

3. This chain can continue as long as:
- Random pops hit pegs of highest value
- Pegs remain on the board to be targeted


Let:

Each Disguise Glasses trigger pops:

Assuming random targeting and uniform peg distribution (this is not the structure of a board, but modeling an arbitrary board at any given round is difficult, so for now this will have to do):

Probability that a given pop hits a highest-value peg:

Expected number of new highest-value peg pops per trigger:

This value determines whether the chain will sustain itself.


To sustain a chain or move toward a full clear:


If the expected number of highest-value peg pops per Disguise Glasses trigger is ≥ 1, the chain is self-sustaining.

Example 1: 2 Unupgraded Glasses, h = 2, r = 16

Example 2: 3 Upgraded Glasses, h = 3, r = 12

• Disguise Glasses are most effective on boards with multiple high-value pegs of the same value.
  
• Their chain potential weakens rapidly as high-value pegs are removed or fall below the top tier.
  
• Chains tend to "fizzle out" as the board becomes sparse or unevenly distributed.
  
• Upgraded copies are significantly stronger, as each provides twice the chance of recursion.
  
• Even in non-infinite scenarios, stacked Glasses can rapidly thin out the board and contribute toward full pops and restock cycles.

This equation defines the condition for expected recursion. While it is not a deterministic loop like Pants + House'O'Cards, a well-balanced board and a strong stack of Disguise Glasses can reliably clear large portions of the board and maintain value density in a score-efficient way.

Ideal use cases include:

• Midgame boards with merged values and few or no orphans
  
• Item slot builds focused on reactive chains and pop synergies
  
• Boards aiming for restock acceleration or combo multiplier setups

This section examines flat point items - those that add fixed score values independent of peg values - and analyzes their effectiveness throughout game progression.

Flat Point Items: Definition and Examples

Flat point items provide score through fixed additions rather than through peg value manipulation. Primary examples include:

- Cheese House: +300 points when Nubby dies (+3000 after round 30)
- Poop Butt: +15 points × number of restocks when Nubby hits their first peg
- Finger Puppet: Points equal to bounce count when Nubby bounces
- Flutty: +3 points for each trigger (every 2 pegs popped)
- Squirmy: +5 points for each trigger (every 5 pegs popped)

Mathematical Model: Early Game Economics

To understand the effectiveness of flat point items, we need to compare their contribution to the round's quota. Let:

- FP(r) = Flat point contribution in round r
- Q(r) = Quota for round r
- E(r) = Effectiveness ratio =
Early Game Analysis (Rounds 0-20)

From the One-Restock Model, we can see quota progression in early rounds:


As shown, a single Cheese House can single-handedly clear quotas in the very early game, with E(r) > 1 indicating the flat points exceed the quota. Even at round 15, one flat point item contributes significantly to quota completion.

Mid-Game Transition (Rounds 21-40)

The effectiveness ratio declines sharply as quotas scale:


Note that at round 30, Cheese House's contribution jumps to 3,000, temporarily boosting its effectiveness, but the exponential quota growth quickly outpaces this increase.

Late Game Collapse (Rounds 41+)

By late game, flat points become increasingly irrelevant:


Even with multiple flat point items, their contribution becomes negligible in the endgame. The E(r) approaches zero as board values and quotas increase exponentially.

The Restock Acceleration Problem

Beyond their diminishing returns, flat point items create a secondary issue: they accelerate restock cycles too quickly without corresponding board-clearing capability.

Consider what happens when flat points trigger early restocks:

1. Premature restock occurs before efficiently popping existing pegs
2. New pegs spawn with base values of 2^(floor(round/5))
3. These merge into higher-value pegs
4. Board value increases, leading to higher quotas next round
5. Flat points become even less effective against these elevated quotas

This creates a "debt spiral" where quota requirements outpace your ability to clear the board.

Mathematical Proof: The Exponential Gap

The fundamental mathematical issue is the mismatch in growth rates:

- Quota growth: Exponential (powers of 2)
- Flat point contribution: Linear or at best quadratic (for cumulative items like Flutty)

Specifically, board value in round r under the One-Restock Model approaches:

Where C₁ is a constant based on board composition.

Meanwhile, flat point contribution scales at best as:

For any positive constants C₁ and C₂, there exists a round number r* such that:

This guarantees that flat points eventually become irrelevant, regardless of how many you stack.

Strategic Implications

Despite their late-game falloff, flat point items serve important strategic roles:

1. Early Economy Building: They can help you survive the first 20 rounds with minimal board-clearing ability
2. Transition Resource: The extra coins from early restocks let you purchase more powerful items
3. Supplemental Scoring: They can provide the "last push" to reach quota when your board-clearing falls just short


The optimal approach is a transition strategy:

- Rounds 0-20: Utilize flat point items to build economy while developing your long-term plan.
- Rounds 30-40: Finish transitioning to your main strategy and perk loadout.
- Rounds 40+: Phase out and sell flat point items in favor of multipliers and loop enablers

Conclusion

Flat point items follow the same mathematical pattern as many game resources: powerful early, negligible late. Even in the pessimistic One-Restock Model, they serve a critical function in jump-starting your economy before recursive strategies become viable.

The key insight: flat points are tools for acceleration, not sustenance. Recognize their declining effectiveness curve and plan your transition to sustainable pop loops before reaching the "flat point cliff" around rounds 30-40.

This section examines a powerful but temporary economic exploit I have termed the "YIMBY Combo" that leverages specific perk-item interactions to generate unprecedented restock chains early in the game.


1. Cheese House (item): Provides +300 points when Nubby dies (+3000 after round 30). Poop Butt also works in this role, but less efficiently, in exchange for being actually useful after this round.
2. Trophy (perk): 50% chance to force-trigger the item in slot 3 when passing round goal


The YIMBY Combo works because of a critical recursive interaction:

- Each "pass round goal" event gives Trophy a 50% chance to trigger
- When Trophy triggers, it force-activates Cheese House
- Force-activated Cheese House gives +300/+3000 points
- These points can immediately trigger another quota clear
- Each new quota clear gives another 50% chance for Trophy to trigger

This creates a probabilistic chain reaction that can generate multiple restocks in rapid succession.


Let's define:
P = 0.5: probability of Trophy triggering on a single quota clear
Q(r): Quota in round r
CH = 300: Points from the Cheese House.

The critical insight: each Cheese House activation can trigger multiple quota clears, and each quota clear gives Trophy an independent chance to trigger.

For example, at Round 9 with a quota of 88:
- Cheese House gives +300 points
- This clears the quota 3 times (⌊300 ÷ 88⌋ = 3)
- Each quota clear gives Trophy a 50% chance to trigger
- Probability of at least one trigger in 3 attempts = (87.5%)

With an 87.5% chance of continuation, the chain becomes explosive:


The expected number of Cheese House activations follows:

With p = 0.875, E(activations) = 8, but this is a conservative estimate because each activation can trigger multiple new activations, causing the chain to have an increasing probability of continuation as it accelerates.

This creates an exponential cascade that can easily reach hundreds or thousands of restocks, effectively maxing out at the game's internal restock limit of 9,999, though practical observations show 300-400 restocks being common.


The YIMBY Combo has two primary viable windows:

Window 1: Round 9
From the One-Restock Model, we know the quota at Round 9 is approximately 88 points.
Cheese House provides 300 points, which is 3.4× the quota.

For the combo to chain:
- First restock occurs naturally
- With 300 points and quota of 88, you clear the quota 3 times
- Each quota clear gives Trophy a 50% chance to trigger
- If any of these trigger, Cheese House activates again
- Each new +300 creates 3 more quota clears
- Each of these gives more chances for Trophy to trigger
- This creates an exponential cascade that can reach hundreds if not thousands of restocks

Window 2: Round 30
At Round 30, the quota is approximately 3,341 points.
Cheese House now provides 3,000 points (.9x of quota), while an upgraded Cheese House provides 7,500 points (2.24x of quota).
At this point, you may also have multiple Trophy Perks.

This creates two scenarios:
1. If your score was already near quota, Cheese House can push you over
2. If you've accumulated even a small score buffer, each activation can trigger another restock


Given the potential for hundreds of restocks, let's model a more realistic YIMBY chain at Round 9:


Total: +300 coins (effectively maxing out at 99), board value equivalent to round 30 or so.

This represents a massive economic discontinuity, essentially skipping 20 rounds of progression in a single turn.


The immediate consequence is a catastrophic jump in quota for the following round:


This represents a 50-100× increase in quota requirements, which renders flat point items completely irrelevant overnight. Even Cheese House's +300 becomes less than 6% of a single quota clear, effectively forcing an immediate transition to mid-game strategies.


The rapid merges create a significantly different board composition:
- A few extremely high-value pegs, surrounded by
- Many relatively low-value pegs
- Higher value variance
- More challenging board to clear efficiently


The YIMBY Combo creates a "one-time economic boost" with specific trade-offs:

Advantages:
- Massive early-game coin acceleration
- Potential to hit the 99 coin cap extremely early
- Access to the cafe and items much earlier, allowing you to go all in on Strawberry and Kebab strategies ASAP

Disadvantages:
- Immediate obsolescence of flat point strategies
- Forces transition to value-based strategies early
- Creates extreme quota pressure in subsequent rounds


The minimum requirements for attempting the YIMBY Combo are:
- Reaching Round 9 with Cheese House in slot 3
- Obtaining Trophy as your first perk
- Having a low enough quota that Cheese House will clear it at least twice, but ideally more than that. The lower, the better.

Additionally, the round 30 variation is an excellent way to rescue a flagging economy should you have the tools to access it and a low enough quota.


This combo demonstrates the ultimate form of flat point exploitation - using a single flat point item to create such an overwhelming economic advantage that flat points themselves become obsolete. It's the mathematical equivalent of burning the ladder you climbed up on, forcing an immediate transition to entirely different strategies.

The YIMBY is less a sustainable strategy and more an all-in economic reset button that catapults the player from early to mid-game in a single round. Players should decide whether they want to experience the game's natural progression curve or accelerate directly to mid-game conditions with maxed economy but exponentially harder quotas.

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