Counterexamples: where each rule stops, and where it starts again

Twenty four rules that hold most of the time. Each one is held against the smallest case that breaks it, and then against the condition that puts it back, so the reading is about the scope of the rule rather than about the rule being false.

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Every part held, and the whole did not

Operations
Pairwise overlap, empty core
The rule as stated
If every pair of sets intersects, all sets share an element.
The smallest thing that breaks it
S1
{a, b}
S2
{b, c}
S3
{a, c}
Pair intersections and common core
{b} · {c} · {a} · ∅
Two sets cannot break the claim because their pair intersection is already their common intersection. Three sets on three elements are minimal.
Where it starts again
A finite family of closed intervals on the real line.
Pairwise intersection implies a common intersection for a finite family of intervals on the real line.
Worked on this page
Mathematics
Every pair fits, the cycle does not
The rule as stated
If every pair of parity constraints is satisfiable, all three are jointly satisfiable.
The smallest thing that breaks it
Constraints
x xor y = 0 · y xor z = 0 · x xor z = 1
Models for each pair
2 · 2 · 2
Cycle xor
1
Joint models
0
A three edge cycle is the smallest closed parity consistency check. Every one or two edge subset is satisfiable.
Where it starts again
Every closed cycle has xor parity zero. Forests satisfy the condition vacuously.
A parity labelled graph is consistent when every closed cycle has xor parity zero.
Worked on this page
Logistics
Balanced arrows, separate circuits
The rule as stated
Equal indegree and outdegree at every vertex guarantees one directed Euler circuit.
The smallest thing that breaks it
Degree at each vertex
a: 1 in, 1 out · b: 1 in, 1 out · c: 1 in, 1 out · d: 1 in, 1 out
Weak components
{a, b} · {c, d}
Closed trails
a → b → a · c → d → c
One Euler circuit
no
With self loops excluded, each nontrivial balanced component needs two vertices. Two disconnected balanced components therefore need four vertices.
Where it starts again
All nonzero degree vertices are connected in the underlying undirected graph and every vertex has equal indegree and outdegree.
Balanced degrees together with one connected nonzero degree component are the condition under which a directed Euler circuit exists.
Named resultEuler circuit, named for Leonhard Euler
Statistics
Every pair is independent, the trio is not
The rule as stated
Pairwise independence implies joint independence.
The smallest thing that breaks it
Equally likely triples
000 · 011 · 101 · 110
Probability of each triple
1/4
Every pair independent
yes
All three jointly independent
no
At least 3 variables are needed to separate pairwise from mutual independence, and 2 states are the smallest nondegenerate state space.
Where it starts again
There are exactly 2 random variables, or full joint factorization is separately established.
For exactly two variables, pairwise independence and joint independence coincide.
Worked on this page
Operations
Two increments become one
The rule as stated
Two concurrent read add write increments must have the same result as two serial increments.
The smallest thing that breaks it
Operations
read · read · add · add · write · write
Serial result
2
Shared result
1
Lost increment
1
One worker cannot overwrite another update. Two workers and one shared location are the smallest lost update construction.
Where it starts again
Each complete update is atomic or serialized by a correct lock, so the shared counter is linearizable.
Two atomic read modify write additions, or two correctly locked updates, produce the serial sum.
SourceLinearizability: A Correctness Condition for Concurrent Objects, ACM, 1990
Scheduling
Every job fits alone, the set does not
The rule as stated
If every job can meet its deadline alone, all jobs can meet their deadlines on one machine.
The smallest thing that breaks it
Jobs
A: 1 by 1 · B: 1 by 1
Each fits alone
yes
Second completion
2 against deadline 1
The set fits
no
One job feasible alone is feasible as the whole set. Two such jobs are enough to exceed a shared deadline.
Where it starts again
Jobs are finite, nonpreemptive, released at time 0, and every earliest deadline prefix has cumulative processing time at most its deadline.
Under those conditions, earliest deadline order is feasible.
Worked on this page
Operations
Majority choices form a circle
The rule as stated
Pairwise majority preference is transitive.
The smallest thing that breaks it
Voter rankings
A > B > C · B > C > A · C > A > B
A > B
2 to 1
B > C
2 to 1
C > A
2 to 1
Majority relation
A → B → C → A
A strict pairwise majority cycle needs at least 3 options, and this strict odd majority construction needs at least 3 voters.
Where it starts again
A finite odd electorate has preferences that are single peaked on one shared axis.
On that restricted preference domain, pairwise majority ordering is transitive under the classical theorem conditions.
Named resultCondorcet cycle, named for Nicolas de Condorcet, and Black single peaked result, named for Duncan Black
Operations
Stable blocks, unstable loop
The rule as stated
Connecting the declared stable scalar first order dynamic block and stable static gain in positive feedback preserves stability.
The smallest thing that breaks it
Dynamic block
G(s) = 2/(s + 1)
Block pole
-1
Loop product
2
Closed loop pole
1
One stable first order dynamic block and one stable static gain form the smallest positive feedback interconnection in the declared model.
Where it starts again
For the declared first order positive feedback family, tau is positive, gains are nonnegative, and k1 times k2 is less than 1.
Under those conditions the closed loop pole is negative and the loop is stable.
Worked on this page

The summary agreed, and what it summarised did not

Each record here holds a summary fixed and changes the case underneath it. The sharpest instance is not on this shelf: two laws with the same mean and the same variance and very different worst days are worked in full, with exact probabilities, on the same moments lesson.

Operations
Row and column totals, no single assignment
The rule as stated
A nonnegative matrix whose rows and columns sum to one is a single assignment, meaning a zero or one permutation matrix.
The smallest thing that breaks it
Matrix
[0.5, 0.5] · [0.5, 0.5]
Row sums
1 · 1
Column sums
1 · 1
Zero or one permutation matrix
no
A 1 by 1 doubly stochastic matrix can only be [1]. Dimension 2 is the first that permits a strict fractional mixture.
Where it starts again
The entries are restricted to 0 or 1, equivalently the matrix is an integral extreme point.
A doubly stochastic matrix with entries restricted to zero or one is a permutation matrix.
SourceOn the Birkhoff von Neumann theorem, Elsevier, 2018
Statistics
Conditioning ties independent causes
The rule as stated
Independent X and Y remain independent after conditioning on the positive probability event Z equals 1, where Z is their OR.
The smallest thing that breaks it
Conditioning event probability
3/4
P(X = 1, Y = 1 | Z = 1)
1/3
Product of conditioned margins
4/9
Independent after conditioning
no
Two binary inputs and one binary conditioning event are the smallest nondegenerate collider construction.
Where it starts again
The conditioning event is independent of the joint pair, or conditional independence is otherwise established in the declared joint model.
Conditioning on an event independent of the pair preserves their joint distribution and independence.
SourceCausal diagrams for empirical research, Oxford University Press, 1995
Statistics
Same margins, different joint tables
The rule as stated
Row and column margins determine the joint table.
The smallest thing that breaks it
Row and column margins
1/2 · 1/2
First joint table
[0.25, 0.25] · [0.25, 0.25]
First determinant
0
Second joint table
[0.5, 0] · [0, 0.5]
Second determinant
0.25
A 2 by 2 table is the smallest table whose fixed margins leave one joint cell free.
Where it starts again
Independence is declared, or another coupling is supplied in addition to the margins.
Under independence, the joint 1,1 cell is r times c and the margins determine all four cells.
Worked on this page

A check at one point did not settle the range

Every record in this group states a result that came out of a finite set of checks. The instrument that reads a sweep, and the reason a clean boundary can sit either side of a band nobody sampled, is worked in full on the unchanged region lesson.

Mathematics

The endpoints stay under, the middle escapes

The rule as stated

If a function is at or below a cap at both endpoints, it stays below the cap between them.

The smallest thing that breaks it

f(0): 1
f(1): 1
f(0.5): 2
Midpoint excess: 1

Two endpoint observations state the premise. One interior observation is the smallest addition that can break it.

Where it starts again

The function is real valued and convex on a closed real interval.

For a convex function on an interval, the maximum does not exceed the larger endpoint value.

SourceConvex Optimization, Cambridge University Press, 2004

Move the assumption

The interior points checked between the endpoints

points

Frompoints

Topoints

In steps ofpoints

It does not check the values in between.

No other integer in the range leaves whether the cap holds across every checked point at the cap holds at every checked point. Every admissible integer in this range was checked.

The answer	the cap holds at every checked point
Values sampled	0 points to 8 points, 9 values, step 1 point
Holds from	0 points, lowest sampled
Holds to	0 points
Nearest value that changes it	1 point
The answer there	a checked interior point exceeds the cap

What an unchanged region does and does not prove is set out in a short lesson.

Every admissible integer in this range was checked. Values outside the range remain unchecked.

Mathematics
No sign change, root still present
The rule as stated
If a continuous function has the same nonzero sign at both ends of an interval, it has no root inside.
The smallest thing that breaks it
Interval
[0, 2]
Endpoint values
1 · 1
Interior root
1 · multiplicity 2
Two endpoint observations and one interior point are the smallest observation pattern. Multiplicity 2 is the lowest even multiplicity that touches without crossing.
Where it starts again
The function is continuous and strictly monotone on the declared closed interval.
For a continuous strictly monotone function, same sign nonzero endpoint values exclude an interior root.
Worked on this page
Mathematics
Newton returns to where it started
The rule as stated
Newton iteration converges whenever a real root exists.
The smallest thing that breaks it
Function
x³ − 2x + 2
Iteration
0 → 1 → 0
Cycle period
2
A nontrivial 2 cycle is the shortest return orbit that is not a fixed point.
Where it starts again
The start is sufficiently close to a simple root and the standard local smoothness and nonzero derivative hypotheses hold.
Newton iteration converges locally under those verified hypotheses.
Named resultNewton method, named for Isaac Newton

Adding something made it worse

Statistics
Better voters, worse majority
The rule as stated
A three voter majority is at least as accurate as each voter when each is better than chance.
The smallest thing that breaks it
Each voter accuracy
3/5
Correlated majority accuracy
2/5
Independent majority at the same accuracy
81/125
Majority below chance
yes
A strict odd majority that can differ from an individual needs 3 voters. The witness uses that smallest panel.
Where it starts again
The 3 voters are independent and have one common accuracy q strictly between 1/2 and 1.
For three independent voters with common accuracy q above one half, majority accuracy is 3q squared minus 2q cubed and exceeds q.
Worked on this page
Operations
A third option reverses two scores
The rule as stated
Under criterion wise minimum and maximum normalization over the current option set, adding an option dominated by existing options cannot reverse their weighted scores.
The smallest thing that breaks it
Weights
3/5 · 2/5
Before C
A 3/5 · B 2/5 · A > B
After C
A 35/55 · B 52/55 · B > A
Normalization
minimum and maximum of the current options
Three alternatives and 2 criteria are the smallest setting in which a new alternative can move normalization anchors and reverse an existing pair.
Where it starts again
Criterion anchors, benefit directions, and weights are fixed externally and do not depend on the current alternative set.
With fixed external anchors and fixed weights, adding an alternative leaves existing normalized values and arithmetic scores unchanged.
SourceOn a short coming of Saaty's method of analytic hierarchies, Elsevier, 1983 Rank reversal and normalization in multi criteria decision making, Wiley, 2021
Operations
A backup adds a point of failure
The rule as stated
Adding a parallel backup together with a required shared series switch always raises mission reliability.
The smallest thing that breaks it
Unit and switch reliability
0.9 · 0.9
One unit
0.9
Two units before the switch
0.99
Two units with the switch
0.891
Reliability loss
0.009
Two parallel units plus one required shared series switch are the smallest architecture that adds a backup and a new common failure point.
Where it starts again
The switch is perfect, unit failures are independent, and no other series failure mode is added.
With a perfect switch and independent unit failures, 2 parallel units have reliability 1 minus (1 minus R) squared, which is greater than R for 0 less than R less than 1.
Worked on this page

Logistics

A new link raises every route time

The rule as stated

In a nonatomic user equilibrium, adding a zero travel time cross link cannot increase equilibrium travel time.

The smallest thing that breaks it

Demand and fixed edge time: 4000 · 45
Without the cross link: 65 minutes
With the cross link: 80 minutes
Alternative path times: 85 · 85
Increase: 15 minutes

Two parallel source to sink routes plus one cross link are the smallest topology in the declared symmetric Braess family.

Where it starts again

Existing route flows are held fixed, or a separately defined system optimum treats the new link as optional.

With fixed route flows the old feasible flow remains available. Under a system optimum, expanding the feasible set cannot worsen the optimum.

Named resultBraess paradox, named for Dietrich Braess

Move the assumption

The travel time of the new link

minutes

Fromminutes

Tominutes

In steps ofminutes

It does not check the values in between.

Across the values sampled, which routing is faster stays the routing without the link is faster from 0 minutes, the lowest value sampled to 20 minutes. The nearest value where it changes is 25 minutes, where it becomes both routings take the same time.

The answer	the routing without the link is faster
Values sampled	0 minutes to 30 minutes, 7 values, step 5 minutes
Holds from	0 minutes, lowest sampled
Holds to	20 minutes
Nearest value that changes it	25 minutes
The answer there	both routings take the same time

What an unchanged region does and does not prove is set out in a short lesson.

The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

The best step each time did not reach the best end

Logistics

The largest coin takes more pieces

The rule as stated

Repeatedly taking the largest usable coin minimizes the number of coins.

The smallest thing that breaks it

Denominations: 1 · 3 · 4
First failure in the entered scan: 6
Largest first: 4 + 1 + 1 · 3 coins
Fewest coins: 3 + 3 · 2 coins

The implementation certifies amount 6 as the first failure for denominations 1, 3, 4 within the entered scan. It makes no global minimality claim across other denomination systems.

Where it starts again

The denomination system has been independently proved canonical. A sufficient special case is that every denomination divides the next.

Largest first greedy uses the fewest coins on a denomination system independently proved canonical.

Worked on this page

Move the assumption

The amount being made up

units

Fromunits

Tounits

In steps ofunits

It does not check the values in between.

No other integer in the range leaves whether largest first uses the fewest coins at largest first uses more coins. Every admissible integer in this range was checked.

The answer	largest first uses more coins
Values sampled	1 unit to 30 units, 30 values, step 1 unit
Holds from	6 units
Holds to	6 units
Nearest value that changes it	5 units and 7 units
The answer there	largest first uses the fewest coins

What an unchanged region does and does not prove is set out in a short lesson.

Every admissible integer in this range was checked. Values outside the range remain unchecked.

Scheduling
The shortest interval blocks more work
The rule as stated
Repeatedly selecting the shortest compatible interval maximizes the number selected.
The smallest thing that breaks it
Intervals
A [0, 3) · B [2, 4) · C [3, 6)
Shortest first
B · 1
Compatible alternative
A, C · 2
Laminar family
no
Deleting any one interval removes the gap. With at most 2 intervals, shortest first cannot select fewer than the largest compatible set.
Where it starts again
The interval family is laminar. For unrestricted unweighted intervals it is the earliest finish rule that holds, not the shortest duration one.
Shortest duration greedy is valid on laminar interval families, while earliest finish is valid for unrestricted unweighted intervals.
Worked on this page
Logistics
A settled shortest path changes later
The rule as stated
Dijkstra settled target result remains correct with a negative edge when there is no negative cycle.
The smallest thing that breaks it
Edges
s → t: 1 · s → a: 2 · a → t: −2
Settled target
1
Later detour
2 + (−2) = 0
Negative cycle
no
Three vertices are necessary to settle a direct target before discovering a cheaper 2 edge detour. The graph is acyclic.
Where it starts again
The graph is finite and every edge weight is nonnegative.
A settled Dijkstra distance is final on a finite graph when every edge weight is nonnegative.
Named resultDijkstra shortest path algorithm, named for Edsger Dijkstra

The statement was exact, and the machine was not

Mathematics
Addition order changes the sum
The rule as stated
Binary64 addition gives the same result regardless of parenthesization.
The smallest thing that breaks it
Operands
2^53 · 1 · −2^53
(a + b) + c
0
a + (b + c)
1
Exact sum
1
Associativity compares 2 parenthesizations of 3 operands. Removing any operand leaves no associativity comparison.
Where it starts again
The arithmetic is exact integer or rational, or the operands are safe integers whose intermediate sums stay safe under both groupings.
Addition is associative in exact arithmetic and for integer operands when every intermediate sum in both permitted groupings stays in the safe integer range.
Worked on this page

Mathematics

A smaller step gives a worse derivative

The rule as stated

A smaller binary64 forward difference step for exp at x equals 1 always gives a more accurate derivative.

The smallest thing that breaks it

Larger step: 2^-26
Larger step error: 3.666088899123565e-8
Smaller step: 2^-54
Input changed: no
Smaller step error: 2.718281828459045

Two distinct step sizes are the smallest comparison that can reverse a monotone accuracy claim.

Where it starts again

In binary64 the step must change the input and preserve a resolvable subtraction, with truncation error and rounding error balanced. In exact real arithmetic the claim is only a limiting statement.

In exact real arithmetic the forward quotient of exp approaches the derivative as a positive step tends to zero.

Worked on this page

Move the assumption

The step size as a power of two

power of two

Frompower of two

Topower of two

In steps ofpower of two

It does not check the values in between.

Across the values sampled, which step gives the closer derivative stays the fixed larger step is closer from -54 power of two, the lowest value sampled to -30 power of two. The nearest value where it changes is -28 power of two, where it becomes the two steps have the same error.

The answer	the fixed larger step is closer
Values sampled	-54 power of two to -24 power of two, 16 values, step 2 power of two
Holds from	-54 power of two, lowest sampled
Holds to	-30 power of two
Nearest value that changes it	-28 power of two
The answer there	the two steps have the same error

What an unchanged region does and does not prove is set out in a short lesson.

The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

Mathematics
The quadratic formula loses a root
The rule as stated
The textbook quadratic formula computes both real roots with comparable relative accuracy in binary64.
The smallest thing that breaks it
Polynomial
x² + 2^28x + 1
Naive roots
0 · -268435456
Stable roots
-268435456 · -3.725290298461914e-9
Product residuals
-1 · 0
A quadratic is the lowest degree polynomial with 2 formula branches, and this coefficient family isolates cancellation of nearly equal binary64 values.
Where it starts again
The discriminant, cancellation aware q branch, and scaled verification remain in range, with a separate repeated root branch.
When q is nonzero, the cancellation aware roots are q/a and c/q, where q equals −0.5 times (b + sign(b) times sqrt(discriminant)).
SourceWhat Every Computer Scientist Should Know About Floating Point Arithmetic, ACM, 1991

Counterexamples: where each rule stops, and where it starts again

Every part held, and the whole did not

Pairwise overlap, empty core

Operations

The rule as stated

If every pair of sets intersects, all sets share an element.

The smallest thing that breaks it

S1 {a, b}

S2 {b, c}

S3 {a, c}

Pair intersections and common core {b} · {c} · {a} · ∅

Two sets cannot break the claim because their pair intersection is already their common intersection. Three sets on three elements are minimal.

Where it starts again

A finite family of closed intervals on the real line.

Pairwise intersection implies a common intersection for a finite family of intervals on the real line.

Worked on this page

Every pair fits, the cycle does not

Mathematics

The rule as stated

If every pair of parity constraints is satisfiable, all three are jointly satisfiable.

The smallest thing that breaks it

Constraints x xor y = 0 · y xor z = 0 · x xor z = 1

Models for each pair 2 · 2 · 2

Cycle xor 1

Joint models 0

A three edge cycle is the smallest closed parity consistency check. Every one or two edge subset is satisfiable.

Where it starts again

Every closed cycle has xor parity zero. Forests satisfy the condition vacuously.

A parity labelled graph is consistent when every closed cycle has xor parity zero.

Worked on this page

Balanced arrows, separate circuits

Logistics

The rule as stated

Equal indegree and outdegree at every vertex guarantees one directed Euler circuit.

The smallest thing that breaks it

Degree at each vertex a: 1 in, 1 out · b: 1 in, 1 out · c: 1 in, 1 out · d: 1 in, 1 out

Weak components {a, b} · {c, d}

Closed trails a → b → a · c → d → c

One Euler circuit no

With self loops excluded, each nontrivial balanced component needs two vertices. Two disconnected balanced components therefore need four vertices.

Where it starts again

All nonzero degree vertices are connected in the underlying undirected graph and every vertex has equal indegree and outdegree.

Balanced degrees together with one connected nonzero degree component are the condition under which a directed Euler circuit exists.

Named result Euler circuit, named for Leonhard Euler

Every pair is independent, the trio is not

Statistics

The rule as stated

Pairwise independence implies joint independence.

The smallest thing that breaks it

Equally likely triples 000 · 011 · 101 · 110

Probability of each triple 1/4

Every pair independent yes

All three jointly independent no

At least 3 variables are needed to separate pairwise from mutual independence, and 2 states are the smallest nondegenerate state space.

Where it starts again

There are exactly 2 random variables, or full joint factorization is separately established.

For exactly two variables, pairwise independence and joint independence coincide.

Worked on this page

Two increments become one

Operations

The rule as stated

Two concurrent read add write increments must have the same result as two serial increments.

The smallest thing that breaks it

Operations read · read · add · add · write · write

Serial result 2

Shared result 1

Lost increment 1

One worker cannot overwrite another update. Two workers and one shared location are the smallest lost update construction.

Where it starts again

Each complete update is atomic or serialized by a correct lock, so the shared counter is linearizable.

Two atomic read modify write additions, or two correctly locked updates, produce the serial sum.

Source Linearizability: A Correctness Condition for Concurrent Objects, ACM, 1990

Every job fits alone, the set does not

Scheduling

The rule as stated

If every job can meet its deadline alone, all jobs can meet their deadlines on one machine.

The smallest thing that breaks it

Jobs A: 1 by 1 · B: 1 by 1

Each fits alone yes

Second completion 2 against deadline 1

The set fits no

One job feasible alone is feasible as the whole set. Two such jobs are enough to exceed a shared deadline.

Where it starts again

Jobs are finite, nonpreemptive, released at time 0, and every earliest deadline prefix has cumulative processing time at most its deadline.

Under those conditions, earliest deadline order is feasible.

Worked on this page

Majority choices form a circle

Operations

The rule as stated

Pairwise majority preference is transitive.

The smallest thing that breaks it

Voter rankings A > B > C · B > C > A · C > A > B

A > B 2 to 1

B > C 2 to 1

C > A 2 to 1

Majority relation A → B → C → A

A strict pairwise majority cycle needs at least 3 options, and this strict odd majority construction needs at least 3 voters.

Where it starts again

A finite odd electorate has preferences that are single peaked on one shared axis.

On that restricted preference domain, pairwise majority ordering is transitive under the classical theorem conditions.

Named result Condorcet cycle, named for Nicolas de Condorcet, and Black single peaked result, named for Duncan Black

Stable blocks, unstable loop

Operations

The rule as stated

Connecting the declared stable scalar first order dynamic block and stable static gain in positive feedback preserves stability.

The smallest thing that breaks it

Dynamic block G(s) = 2/(s + 1)

Block pole -1

Loop product 2

Closed loop pole 1

One stable first order dynamic block and one stable static gain form the smallest positive feedback interconnection in the declared model.

Where it starts again

For the declared first order positive feedback family, tau is positive, gains are nonnegative, and k1 times k2 is less than 1.

Under those conditions the closed loop pole is negative and the loop is stable.

Worked on this page

The summary agreed, and what it summarised did not

Row and column totals, no single assignment

Operations

The rule as stated

A nonnegative matrix whose rows and columns sum to one is a single assignment, meaning a zero or one permutation matrix.

The smallest thing that breaks it

Matrix [0.5, 0.5] · [0.5, 0.5]

Row sums 1 · 1

Column sums 1 · 1

Zero or one permutation matrix no

A 1 by 1 doubly stochastic matrix can only be [1]. Dimension 2 is the first that permits a strict fractional mixture.

Where it starts again

The entries are restricted to 0 or 1, equivalently the matrix is an integral extreme point.

A doubly stochastic matrix with entries restricted to zero or one is a permutation matrix.

Source On the Birkhoff von Neumann theorem, Elsevier, 2018

Conditioning ties independent causes

Statistics

The rule as stated

Independent X and Y remain independent after conditioning on the positive probability event Z equals 1, where Z is their OR.

The smallest thing that breaks it

Conditioning event probability 3/4

P(X = 1, Y = 1 | Z = 1) 1/3

Product of conditioned margins 4/9

Independent after conditioning no

Two binary inputs and one binary conditioning event are the smallest nondegenerate collider construction.

Where it starts again

The conditioning event is independent of the joint pair, or conditional independence is otherwise established in the declared joint model.

Conditioning on an event independent of the pair preserves their joint distribution and independence.

Source Causal diagrams for empirical research, Oxford University Press, 1995

Same margins, different joint tables

Statistics

The rule as stated

Row and column margins determine the joint table.

The smallest thing that breaks it

Row and column margins 1/2 · 1/2

First joint table [0.25, 0.25] · [0.25, 0.25]

First determinant 0

Second joint table [0.5, 0] · [0, 0.5]

Second determinant 0.25

A 2 by 2 table is the smallest table whose fixed margins leave one joint cell free.

Where it starts again

Independence is declared, or another coupling is supplied in addition to the margins.

Under independence, the joint 1,1 cell is r times c and the margins determine all four cells.

Worked on this page

A check at one point did not settle the range

The endpoints stay under, the middle escapes

Mathematics

The rule as stated

If a function is at or below a cap at both endpoints, it stays below the cap between them.

The smallest thing that breaks it

f(0) 1

f(1) 1

f(0.5) 2

Midpoint excess 1

Two endpoint observations state the premise. One interior observation is the smallest addition that can break it.

Where it starts again

The function is real valued and convex on a closed real interval.

For a convex function on an interval, the maximum does not exceed the larger endpoint value.

Source Convex Optimization, Cambridge University Press, 2004

Sweep of The interior points checked between the endpoints

The assumption: The interior points checked between the endpoints
Values sampled: 0 to 8, 9, 1
The answer: the cap holds at every checked point
Holds across: 0 to 0
Nearest change: 1
Reading the boundary: The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

No sign change, root still present

Mathematics

The rule as stated

If a continuous function has the same nonzero sign at both ends of an interval, it has no root inside.

The smallest thing that breaks it

Interval [0, 2]

Endpoint values 1 · 1

Interior root 1 · multiplicity 2

Two endpoint observations and one interior point are the smallest observation pattern. Multiplicity 2 is the lowest even multiplicity that touches without crossing.

Where it starts again

The function is continuous and strictly monotone on the declared closed interval.

For a continuous strictly monotone function, same sign nonzero endpoint values exclude an interior root.

Worked on this page

Newton returns to where it started

Mathematics

The rule as stated

Newton iteration converges whenever a real root exists.

The smallest thing that breaks it

Function x³ − 2x + 2

Iteration 0 → 1 → 0

Cycle period 2

A nontrivial 2 cycle is the shortest return orbit that is not a fixed point.

Where it starts again

The start is sufficiently close to a simple root and the standard local smoothness and nonzero derivative hypotheses hold.

Newton iteration converges locally under those verified hypotheses.

Named result Newton method, named for Isaac Newton

Adding something made it worse

Better voters, worse majority

Statistics

The rule as stated

A three voter majority is at least as accurate as each voter when each is better than chance.

The smallest thing that breaks it

Each voter accuracy 3/5

Correlated majority accuracy 2/5

Independent majority at the same accuracy 81/125

Majority below chance yes

A strict odd majority that can differ from an individual needs 3 voters. The witness uses that smallest panel.

Where it starts again

The 3 voters are independent and have one common accuracy q strictly between 1/2 and 1.

For three independent voters with common accuracy q above one half, majority accuracy is 3q squared minus 2q cubed and exceeds q.

Worked on this page

A third option reverses two scores

Operations

The rule as stated

Under criterion wise minimum and maximum normalization over the current option set, adding an option dominated by existing options cannot reverse their weighted scores.

The smallest thing that breaks it

Weights 3/5 · 2/5

Before C A 3/5 · B 2/5 · A > B

After C A 35/55 · B 52/55 · B > A

Normalization minimum and maximum of the current options

Three alternatives and 2 criteria are the smallest setting in which a new alternative can move normalization anchors and reverse an existing pair.

Where it starts again

Criterion anchors, benefit directions, and weights are fixed externally and do not depend on the current alternative set.

With fixed external anchors and fixed weights, adding an alternative leaves existing normalized values and arithmetic scores unchanged.

Source On a short coming of Saaty's method of analytic hierarchies, Elsevier, 1983 Rank reversal and normalization in multi criteria decision making, Wiley, 2021

A backup adds a point of failure

Operations

The rule as stated

Adding a parallel backup together with a required shared series switch always raises mission reliability.

The smallest thing that breaks it

Unit and switch reliability 0.9 · 0.9

One unit 0.9

Two units before the switch 0.99

Two units with the switch 0.891

Reliability loss 0.009

Two parallel units plus one required shared series switch are the smallest architecture that adds a backup and a new common failure point.

Where it starts again

The switch is perfect, unit failures are independent, and no other series failure mode is added.

With a perfect switch and independent unit failures, 2 parallel units have reliability 1 minus (1 minus R) squared, which is greater than R for 0 less than R less than 1.

Worked on this page

A new link raises every route time

Logistics

The rule as stated

In a nonatomic user equilibrium, adding a zero travel time cross link cannot increase equilibrium travel time.

The smallest thing that breaks it

Demand and fixed edge time 4000 · 45

Without the cross link 65 minutes

With the cross link 80 minutes

Alternative path times 85 · 85

Increase 15 minutes

Two parallel source to sink routes plus one cross link are the smallest topology in the declared symmetric Braess family.

Where it starts again

Existing route flows are held fixed, or a separately defined system optimum treats the new link as optional.

With fixed route flows the old feasible flow remains available. Under a system optimum, expanding the feasible set cannot worsen the optimum.

Named result Braess paradox, named for Dietrich Braess

Sweep of The travel time of the new link

The assumption: The travel time of the new link
Values sampled: 0 to 30, 7, 5
The answer: the routing without the link is faster
Holds across: 0 to 20
Nearest change: 25
Reading the boundary: The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

The best step each time did not reach the best end

The largest coin takes more pieces

Logistics

The rule as stated

Repeatedly taking the largest usable coin minimizes the number of coins.

The smallest thing that breaks it

Denominations 1 · 3 · 4

First failure in the entered scan 6

Largest first 4 + 1 + 1 · 3 coins

Fewest coins 3 + 3 · 2 coins

The implementation certifies amount 6 as the first failure for denominations 1, 3, 4 within the entered scan. It makes no global minimality claim across other denomination systems.

Where it starts again

The denomination system has been independently proved canonical. A sufficient special case is that every denomination divides the next.

Largest first greedy uses the fewest coins on a denomination system independently proved canonical.

Worked on this page

Sweep of The amount being made up

The assumption: The amount being made up
Values sampled: 1 to 30, 30, 1
The answer: largest first uses more coins
Holds across: 6 to 6
Nearest change: 5 and 7
Reading the boundary: The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

The shortest interval blocks more work

Scheduling

The rule as stated

Repeatedly selecting the shortest compatible interval maximizes the number selected.

The smallest thing that breaks it

Intervals A [0, 3) · B [2, 4) · C [3, 6)

Shortest first B · 1

Compatible alternative A, C · 2

Laminar family no

Deleting any one interval removes the gap. With at most 2 intervals, shortest first cannot select fewer than the largest compatible set.

Where it starts again

The interval family is laminar. For unrestricted unweighted intervals it is the earliest finish rule that holds, not the shortest duration one.

Shortest duration greedy is valid on laminar interval families, while earliest finish is valid for unrestricted unweighted intervals.

Worked on this page

A settled shortest path changes later

Logistics

The rule as stated

Dijkstra settled target result remains correct with a negative edge when there is no negative cycle.

The smallest thing that breaks it

Edges s → t: 1 · s → a: 2 · a → t: −2

Settled target 1

Later detour 2 + (−2) = 0

Negative cycle no

Three vertices are necessary to settle a direct target before discovering a cheaper 2 edge detour. The graph is acyclic.

Where it starts again

The graph is finite and every edge weight is nonnegative.

A settled Dijkstra distance is final on a finite graph when every edge weight is nonnegative.

Named result Dijkstra shortest path algorithm, named for Edsger Dijkstra

The statement was exact, and the machine was not

Addition order changes the sum

Mathematics

The rule as stated

Binary64 addition gives the same result regardless of parenthesization.

The smallest thing that breaks it

Operands 2^53 · 1 · −2^53

(a + b) + c 0

a + (b + c) 1

Exact sum 1

Associativity compares 2 parenthesizations of 3 operands. Removing any operand leaves no associativity comparison.

Where it starts again

The arithmetic is exact integer or rational, or the operands are safe integers whose intermediate sums stay safe under both groupings.

Addition is associative in exact arithmetic and for integer operands when every intermediate sum in both permitted groupings stays in the safe integer range.

Worked on this page

A smaller step gives a worse derivative

Mathematics

The rule as stated

A smaller binary64 forward difference step for exp at x equals 1 always gives a more accurate derivative.

The smallest thing that breaks it

Larger step 2^-26

Larger step error 3.666088899123565e-8

Smaller step 2^-54

Input changed no

Smaller step error 2.718281828459045

Two distinct step sizes are the smallest comparison that can reverse a monotone accuracy claim.

Where it starts again

In binary64 the step must change the input and preserve a resolvable subtraction, with truncation error and rounding error balanced. In exact real arithmetic the claim is only a limiting statement.

In exact real arithmetic the forward quotient of exp approaches the derivative as a positive step tends to zero.

Worked on this page

Sweep of The step size as a power of two

The assumption: The step size as a power of two
Values sampled: -54 to -24, 16, 2
The answer: the fixed larger step is closer
Holds across: -54 to -30
Nearest change: -28
Reading the boundary: The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

The quadratic formula loses a root

Mathematics

The rule as stated

The textbook quadratic formula computes both real roots with comparable relative accuracy in binary64.

The smallest thing that breaks it

Polynomial x² + 2^28x + 1

Naive roots 0 · -268435456

Stable roots -268435456 · -3.725290298461914e-9

Product residuals -1 · 0

A quadratic is the lowest degree polynomial with 2 formula branches, and this coefficient family isolates cancellation of nearly equal binary64 values.

Where it starts again

The discriminant, cancellation aware q branch, and scaled verification remain in range, with a separate repeated root branch.

When q is nonzero, the cancellation aware roots are q/a and c/q, where q equals −0.5 times (b + sign(b) times sqrt(discriminant)).

Source What Every Computer Scientist Should Know About Floating Point Arithmetic, ACM, 1991

Not on this shelf, and why

The equal moments and different tails record is worked in full on the same moments lesson, so this shelf links to that lesson instead of shortening it into a card.

A rule whose breaker exists only outside the domain a practitioner would work in stays off this shelf because the construction would be true and unusable here.

A rule without a restoration condition stays off this shelf because its scope is not yet understood well enough to describe.

Every part held, and the whole did not

Pairwise overlap, empty core

Every pair fits, the cycle does not

Balanced arrows, separate circuits

Every pair is independent, the trio is not

Two increments become one

Every job fits alone, the set does not

Majority choices form a circle

Stable blocks, unstable loop

The summary agreed, and what it summarised did not

Row and column totals, no single assignment

Conditioning ties independent causes

Same margins, different joint tables

A check at one point did not settle the range

The endpoints stay under, the middle escapes

The interior points checked between the endpoints

No sign change, root still present

Newton returns to where it started

Adding something made it worse

Better voters, worse majority

A third option reverses two scores

A backup adds a point of failure

A new link raises every route time

The travel time of the new link

The best step each time did not reach the best end

The largest coin takes more pieces

The amount being made up

The shortest interval blocks more work

A settled shortest path changes later

The statement was exact, and the machine was not

Addition order changes the sum

A smaller step gives a worse derivative

The step size as a power of two

The quadratic formula loses a root

Every part held, and the whole did not

Pairwise overlap, empty core

Every pair fits, the cycle does not

Balanced arrows, separate circuits

Every pair is independent, the trio is not

Two increments become one

Every job fits alone, the set does not

Majority choices form a circle

Stable blocks, unstable loop

The summary agreed, and what it summarised did not

Row and column totals, no single assignment

Conditioning ties independent causes

Same margins, different joint tables

A check at one point did not settle the range

The endpoints stay under, the middle escapes

Sweep of The interior points checked between the endpoints

No sign change, root still present

Newton returns to where it started

Adding something made it worse

Better voters, worse majority

A third option reverses two scores

A backup adds a point of failure

A new link raises every route time

Sweep of The travel time of the new link

The best step each time did not reach the best end

The largest coin takes more pieces

Sweep of The amount being made up

The shortest interval blocks more work

A settled shortest path changes later

The statement was exact, and the machine was not

Addition order changes the sum

A smaller step gives a worse derivative

Sweep of The step size as a power of two

The quadratic formula loses a root

Not on this shelf, and why

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