MathJewels.com
Problem 1 · The Machine That Repeats ForeverPrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 1

The Machine That Repeats Forever

Student worksheets
1 generic worksheet · no answer keyIf this browser cannot open its printer dialog, the ready-to-print PDF opens automatically.
MathJewels.com
Name
Date

The Machine That Repeats Forever

Complete to collectTumbled SunstoneLevel 2 · 4 Glints

Meditations on Oneness

Local folio mark · 1

Page 1 · Let the machine pass through itself

Rook's endless test

In the Observatory, Rook feeds each output back through the same multiplication machine. The machine accepts any nonnegative whole number, and every pass multiplies its input by the same hidden nonnegative whole number . Write .

Rook can make the chain as long as requested. Your proof must account for every pass.
Reversible

An undo machine recovers every input, so for every .

Repeat-stable

Two passes have exactly the same effect as one: for every .

An Observatory diagram compares one multiplication-machine pass, two chained passes, and a long chain of any positive number r of passes, including 11 and 111. The given rule says two passes have the same output as one, and a gold return arrow represents an undo machine.
One pass and two passes agree. What does that force every longer chain to do?

Follow the repetitions

Use separate paper for your proof.

  1. Without finding , prove that passes have exactly the same output as one pass for every positive whole number . State what this says about 11 passes and 111 passes.

Page 2 · Find the hidden multiplier

Two candidates, one reversible survivor

Now use that is a nonnegative whole number. For the input , repeat-stability says that the one-pass and two-pass outputs agree.

Trace the input through both routes.

Let reversibility decide

Use separate paper for your proof.

  1. Use the input to prove that . Prove that is or by explaining why when . Then rule out : show that the different inputs and would both produce , so one undo machine could not recover both inputs. Conclude that .

The hidden multiplier is

After the first pass, every later pass is only an echo.

At a glance

About this problem

These tags show what the problem is about and the math you’ll use. Choose one to find similar problems.

Theme

Skills

Helpful first

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 6 · Grade 7 · Grade 8

Uses Middle Grade 7 math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 6–13 minutes

Moderate amount of written work · Proof response.

How to support: The useful tension is that zero and one both repeat stably. Ask what collision multiplying by zero creates before the learner chooses between them.

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

Keep exploring

The Manyfold challenge 1 of 22

Technical reference

Rating, skills & curriculum

Designed for Grade 6 · Grade 7 · Grade 8 · Uses Middle Grade 7 math · About 6–13 minutes

Grade 6 · Grade 7 · Grade 8 · Functions and change · Transformations, composition, and inverses

MJ 7.5 · C2 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ 7.5

Grades 6–8

Grade 7 progression · point 5

Likely range
MJ 7.4–7.7
Confidence
82%

Challenge

C2 · Standard

The solver chooses and coordinates familiar methods.

  • Method selection1
  • Reasoning depth2
  • Novelty1
  • Constraint management1
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
6–13 minutes · typically 9
Response
Proof
Known-method steps
8
  • Mechanical execution1
  • Written output2
  • Representation production0
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Reason about repeat-stable iteration

    The learner can use F(F(x)) = F(x) to prove that every positive iterate equals F(x), then combine repeat-stability with a supplied concrete undo or reversibility condition to determine when F must be the identity.

    • Independent mastery
    • 100% focus
    • MJ 7.5 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Evaluate expressions for specified variable values

    The learner can evaluate expressions for specified variable values. This capability is distinct from mere exposure to or isolated use of substitution; operations.

    • Independent readiness
    • Required
    • MJ 6.6 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Practice locations1 location · Representative: Construct viable arguments and critique the reasoning of others.