Teaching-mechanic library

unit_pipeline · math · transformer.composition

Concept: Dimensional analysis

Player does: Chain conversion-factor pipes; joints connect only when units cancel

Breaks: Units are labels you can drop

Units multiply and cancel like numbers, which is exactly how a chain of conversions works.

order_of_ops_forge · math · transformer.trace

Concept: Order of operations

Player does: Predict what the calculator machine outputs for a nested expression

Breaks: Operations always go left to right

Operations follow precedence and grouping, not reading order.

function_factory · math · transformer.function_machine

Concept: What a function is

Player does: Feed inputs, watch outputs, predict the output for a new input

Breaks: A function must be a formula

A function is any rule pairing each input to exactly one output, whether given by a formula, table, or graph.

reverse_factory · math · transformer.inverse

Concept: Inverse functions

Player does: Run the machine backward to recover the input

Breaks: f⁻¹(x) means 1/f(x)

The inverse function f⁻¹ undoes f by swapping inputs and outputs; it is not the reciprocal 1/f(x).

machine_pipeline · math · transformer.composition

Concept: Composition

Player does: Order two machines so the crate exits as the target

Breaks: f(g(x)) = g(f(x))

Function composition is generally not commutative, so f(g(x)) and g(f(x)) usually differ.

input_filter · math · transformer.domain_filter

Concept: Domain & range

Player does: Choose which crates the √ or 1/x machine accepts without jamming

Breaks: Every real number is a valid input

A function's domain excludes inputs that break it, like negatives under a square root or zero in a denominator.

pattern_path · math · transformer.function_machine

Concept: Arithmetic & geometric sequences

Player does: Jump to the stone holding the next term

Breaks: Every sequence adds a constant

An arithmetic sequence adds a constant difference, but a geometric sequence multiplies by a constant ratio instead.

transform_key · math · transformer.geometric

Concept: Rigid transformations & symmetry

Player does: Rotate, reflect, translate the key until it fits the slot

Breaks: A line reflection equals a 180° rotation

A reflection flips a shape's orientation while a 180° rotation preserves it, so the two are not the same transformation.

derivative_machine · math · transformer.function_machine

Concept: Differentiation rules

Player does: The machine turns f into f′; predict its output for a new f

Breaks: d/dx x³ = 3x³

The power rule brings the exponent down as a coefficient and reduces it by one, so d/dx x³ = 3x², not 3x³.

transformation_grid · math · transformer.matrix

Concept: Matrices as transformations

Player does: Pick the 2×2 matrix that turns the rune into the door's shape

Breaks: Matrix multiplication is commutative

Matrix multiplication generally is not commutative, so applying two transformations in a different order gives a different result.

determinant_press · math · transformer.matrix

Concept: Determinant as area scale

Player does: Predict the stamped area after the matrix press

Breaks: The determinant is the sum of the entries

The determinant measures how a matrix scales area (or volume); it is computed from a product-and-difference formula, not a sum of entries.

genome_repair · biology · transformer.encode

Concept: DNA base pairing

Player does: Rebuild the damaged strand by complementary pairing

Breaks: A pairs with G

DNA base pairing is fixed: A pairs with T, and G pairs with C.

rna_assembly · biology · transformer.encode

Concept: Transcription

Player does: Read the template strand and assemble the mRNA (U for T)

Breaks: mRNA is an exact copy of the template strand

mRNA is complementary to the DNA template strand, not an identical copy, and it swaps thymine for uracil.

protein_factory · biology · transformer.encode

Concept: Translation

Player does: Route codons to amino acids with the codon table to build the chain

Breaks: Each nucleotide codes for one amino acid

A group of three nucleotides, a codon, specifies one amino acid, not a single nucleotide.

photosynthesis_recipe · biology · transformer.composition

Concept: Photosynthesis

Player does: Route light, water, CO₂ through the stations to make glucose

Breaks: Plants get their mass from the soil

Most of a plant's mass comes from carbon dioxide fixed by photosynthesis, not from the soil.

respiration_forge · biology · transformer.composition

Concept: Cellular respiration

Player does: Run glucose through glycolysis, Krebs, and the ETC to charge ATP

Breaks: Plants don't do cellular respiration

Plants perform cellular respiration to make ATP just as animals do, in addition to photosynthesis.

state_containers · cs · transformer.trace

Concept: Variables & assignment

Player does: Predict each container's value after the instructions run

Breaks: x = x + 1 is an impossible equation

In code, = is assignment, not algebraic equality, so x = x + 1 replaces x's stored value with one more than itself.

branch_doors · cs · transformer.trace

Concept: Conditionals

Player does: Predict which door the branching world opens for the current state

Breaks: Both branches of an if run

An if statement runs exactly one of its branches, chosen by whether the condition is true or false.

nested_rooms · cs · transformer.trace

Concept: Recursion

Player does: Each room holds a smaller copy; predict the result as base cases return

Breaks: Recursion never stops

Recursion terminates once it reaches a base case, which stops the chain of smaller calls from growing further.

indexed_inventory · cs · transformer.trace

Concept: Arrays & indexing

Player does: Fetch items by slot from a zero-indexed inventory

Breaks: Indexes count from 1 in every language

Many languages index arrays starting at 0, so the first element sits at index 0, not 1.

ordering_conveyor · cs · transformer.trace

Concept: Sorting algorithms

Player does: Predict the conveyor after pass k of bubble or insertion sort

Breaks: Every sort does the same work

Different sorting algorithms compare and move elements in different patterns, so their intermediate states differ.

bucket_router · cs · transformer.function_machine

Concept: Hashing

Player does: Predict each key's bucket from h(k) = k mod m; spot collisions

Breaks: Hashing sorts the keys

A hash function maps keys to buckets for fast lookup; it does not order the keys.

memory_warehouse · cs · transformer.trace

Concept: Caching & locality

Player does: Predict hits and misses as requests arrive

Breaks: Every memory access costs the same

A cache hit is far cheaper than a miss, so locality of access strongly affects real performance.

cipher_door · cs · transformer.encode

Concept: Binary, hex, encodings

Player does: Translate the carving from binary or hex to unlock the door

Breaks: Hex digits only go 0–9

Hexadecimal digits run 0 through F, using A–F to represent values ten through fifteen.

signal_railway · engineering · transformer.trace

Concept: Multiplexers

Player does: Set select lines so the right input track reaches the output

Breaks: Select lines carry the data

Select lines only choose which data input is routed through; they never carry the data themselves.

one_bit_room · engineering · transformer.trace

Concept: Flip-flops

Player does: Predict the stored bit after each clock edge

Breaks: Outputs change whenever inputs change

A flip-flop's stored output changes only at a clock edge, not whenever its inputs change.

conjugation_forge · language · transformer.function_machine

Concept: Verb conjugation

Player does: Feed person + tense; predict the machine's verb form

Breaks: Irregular verbs follow the regular pattern

Irregular verbs conjugate by their own memorized pattern instead of the regular rule.