Bowling world

Forty bowling lanes compare twelve deliveries and twelve full racks with twenty deliveries and ten full racks.

Twelve is not the minimum

The universe that can be counted

A score loses the work

Throughput changes the denominator

Skill does not mean speed

Forty lanes and four objectives

What the experiment found

What remains outside

Twelve is not the minimum

Ordinary ten-pin scoring gives a strike one delivery in each of the first nine frames and two bonus deliveries after a strike in the tenth. Nine strikes followed by an open tenth can finish in eleven deliveries. Nine strikes followed by 0,0 score 240. The scoring optimum and the minimum-delivery history are different objects.

Complete game	Score	Deliveries	Full-rack services	Standing-pin services
Twelve strikes	300	12	12	0
Nine strikes, then 0,0	240	11	10	1
Nine pins then a miss in every frame	90	20	10	10
Five pins, spare, throughout; five-pin fill	150	21	11	10

The accounting starts with a ready rack and ends ready for the next game. It includes the final service and excludes the starting rack. A standing-pin service means retaining the remaining pins for the next delivery; actual machines can use shortened, out-of-range or other special cycles. These are workload categories, not a claim that every machine runs the same movement on every ball. The current playing rules, Rule 2 fix the scoring grammar.

The universe that can be counted

There are 66 ordinary-frame pin-count outcomes and 241 tenth-frame outcomes. Their product, 66⁹ × 241, is 5,726,805,883,325,784,576 complete histories. This count omits fouls, dead balls, rerolls and physical pin identities. It does not count every trajectory, every form of bowling or every way to run a business.

The lab computes the number of histories behind each score, and the shortest and longest delivery counts compatible with it. A forward dynamic program carries the accumulated score and outstanding strike/spare bonuses. An independent backward calculation uses the first two deliveries of an already-built suffix. They agree across all 301 score totals and workload extrema.

This is a useful limit on claims about complexity. An enormous history space need not require visiting every history. The scoring rules allow histories with equivalent futures to be combined. The count is exact because the retained state is sufficient for scoring. It is not sufficient for collision damage, oil transport, customer preference or a mechanic's next repair.

Throughput changes the denominator

Suppose a lane is continuously occupied, a human delivers every thirty seconds, and machine service finishes inside that interval. Temporarily remove changeover and oil downtime. Both perfect and all-open play then produce 120 deliveries per lane-hour. Perfect play completes ten games; open play completes six. Full-rack services rise from sixty to 120 per hour.

At that limit, improving skill does not reduce the number of balls moving through the building per hour. It changes the mixture of machine work and the number of completed scorecards. Below capacity, a fixed demand for games gives the opposite scheduling consequence: the same customers finish earlier, leaving more idle time. Freed capacity is an option, not a sale.

The interactive separates demand measured in player-games from demand measured in occupied lane-hours. It also separates the unit customers want from the unit the operator bills. With fixed occupied time and hourly pricing, more games can be completed without more lane revenue. Food, drink, membership and repeat visits might alter that result; no revenue from those sources is invented here.

Skill does not mean speed

The model has separate controls for strike probability, spare probability, human cadence and ball speed. Better placement need not mean a faster release. The ball-motion study treats motion as a multivariable problem involving the ball, delivery and lane condition.

At equal mass, twelve balls carry less translational kinetic energy than twenty at the same speed. If the twelve are faster by more than √(20/12), about 1.291 times, the order reverses. Twelve releases at 22 mph carry 13.44% more translational energy than twenty at 16 mph. Energy exposure is not damage: spin, impact geometry, damping and material history matter. No monetary wear forecast is derived from that proxy.

There is another state variable: timing. A motor with forty-five seconds of run-on after each activation stays active for 265 seconds under twelve activations twenty seconds apart, but 540 seconds with eighty-second spacing. This post hoc interval-union illustration shows why a cycle count alone need not determine electrical duty. The financial model's per-cycle energy values remain editable assumptions; it does not claim to calibrate that hardware behavior.

Forty lanes and four objectives

The urban and rural buttons load illustrative demand, wage, property-cost and price assumptions. They are not measured descriptions of either place. Changing the label cannot change the scoring rules or mechanics. Replace the assumptions to represent a particular site.

The daily ledger includes annualized equipment ownership, property overhead, labor, standby and active electricity, delivery/full/standing service costs, oiling cost and downtime, and training charged per served player-game. Human preparation can overlap machine service. Capacity is constrained by whichever is slower, plus changeover and maintenance time. The operating model is a steady-state expectation, not an arrival queue or a prediction of individual waiting times. The interface exposes selected coefficients; equipment capital, financing and other ownership defaults are defined in the model source.

Objective	What the search improves	What it can sacrifice
Minimum total daily cost	money spent per open day	customers served, unless a service floor binds
Minimum cost per player-game	spending divided by completed games	total spending, spare capacity, accessibility
Maximum completed games	scorecards completed	spending, encounter duration, machine workload
Maximum operating surplus	modeled receipts less modeled costs	any unpriced value or consequence

The finite search tests 270 policies: three skill assumptions, five active-lane counts, two equipment assumptions, three oil intervals and three human cadences. The default service floor requires serving at least 90% of the specified demand. It reports no feasible solution when that requirement cannot be met. The perfect policy is a counterfactual bound, not a promise that paying for training produces universal 300 games.

Editing strike probability, spare probability or training cost changes the selected skill profile and its ninety policies in the search. The other two profiles retain their preset values. Switching skill presets clears those profile edits; applying a policy from the same profile retains them. Edited machine service times, prices and electrical coefficients apply to both equipment alternatives as common scenario assumptions. Custom profiles are labeled in the search.

The oil interval is especially consequential: less frequent oiling saves money and time by construction because lane-condition feedback is not yet modeled. A winning policy is conditional on that interval being admissible. The equipment alternatives also use invented costs, timings and energy coefficients; the search is not a product recommendation. Arbitrary ticket prices are not optimized against fixed demand.

What the experiment found

The seven sharply specified scoring, workload and accounting predictions held. The eighth found different policies for cost and throughput, but its numerical demand scenarios and service floor were selected after the initial prediction commit. That result is a conditional demonstration with an incompletely specified registration, not an equally strong prospective test.

The central consequence is a change of state and objective. A game creates a world in which 300 has a precise meaning. A business must add demand, time, maintenance and a purpose before it can say better. Perfect scoring does not settle those choices. People may value learning, difficulty, conversation, competition or an unhurried evening; maximizing completed scorecards would assign those values without asking them.

What remains outside

Bookings, parties and queues; leagues sharing pairs of lanes; staffing skills and shift constraints; measured failure and repair distributions; oil pattern evolution and resurfacing; pin and ball replacement; HVAC and climate; buildings, land, taxes, financing alternatives and residual value; food and drink; seasonality, accessibility and the experience people return for all need additional measurements or choices. The model contains editable proxies for some costs, and no simulated mechanism for most of this list.

The next useful experiment is an instrumented operating shift: delivery timestamps, machine-state transitions, actual power, fault/repair times, oiling, bookings, prices and customer time. That would let the present law-based ledger meet a world that can disagree with its assumed coefficients.

Bowling world?

Twelve is not the minimum?

The universe that can be counted?

A score loses the work?

Throughput changes the denominator?

Skill does not mean speed?

Forty lanes and four objectives?

What the experiment found?

What remains outside?

Bowling world

Twelve is not the minimum

The universe that can be counted

A score loses the work

Throughput changes the denominator

Skill does not mean speed

Forty lanes and four objectives

What the experiment found

What remains outside