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Lesson 27.9›Bonus Chapter 28›Lesson 28.1
Bonus puzzle studio · forward and backward reasoning

Close Circular Arithmetic Chains with Inverse Operations

Arrange every operation token around a ring, carry the running value clockwise, and use inverse operations to make the final step return exactly to the starting number.

circular chainsinverse operationsexact divisionbacktrackingoriginal Example 1 + Practice 1
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Your task: arrange the operations and return to 1

Start at 1, use every token once, and finish at 1. Every value along the way must stay a positive whole number. You choose the token order; the running values follow from your calculations.

  1. Learn the ring: Mission 1 uses a four-token mini-loop so you can see how operations and values alternate.
  2. Solve the six-token puzzle: Missions 2–3 teach backward reasoning and trace one worked route. Mission 4 is your own six-token ring. Mission 5 asks what a successful route proves.
  3. Try a longer ring: Missions 6–7 use nine tokens. The extra digit-tour modes are optional challenges with additional rules.
  4. Transfer the method: Mission 8 gives you a different five-token puzzle. Use the same forward-and-backward reasoning with its new tokens.

A closed chain has one global condition

x0 → operation 1 → x1 → ··· → operation n → x0

Every operation is used once. Every intermediate circle must agree with the operation before it. The final operation must return to the original start.

1. Pin the startDo not move or replace the given starting value.
2. Work both waysCalculate forward, then use inverse operations from the final target.
3. Protect divisionIn the core activities, every division must give a positive whole number.
4. Verify closureA locally correct chain is unfinished until the final value equals the start.

Which values are allowed?

Use positive whole numbers throughout the main puzzles. A division must be exact: 12 ÷ 4 = 3 is allowed, but 5 ÷ 2 = 2.5 is not. Zero and negative results are not allowed.

A separate, clearly labeled digit-tour extension asks for distinct one-digit results. Count the start and the values before the final return; do not count the final repeated start again. In a six-token tour, six different digits are enough; a nine-token tour uses all nine digits 1–9. Choose a digit-tour mode only when you want that extra challenge; ordinary closure does not require one-digit or distinct values.

Puzzle habit: a failed order is evidence. Record the first illegal or unhelpful step, undo only what caused it, and try a better branch.
1

Read a closed operation ring

Operations and running values alternate around the circle
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Trace the mini-loop

Operations4
Steps revealed0 / 4
Final result—
Closed?waiting
2

Reverse one operation to find the previous value

The final target can eliminate impossible last steps
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Now use the six-token puzzle. Work backward from the required final value 1. This activity tests possible last steps; it does not arrange all six tokens yet.

Inverse-operation machine

? ÷ 6 = 1
Required previous value6
Positive whole?yes
Inverse move× 6
Target1

Required inputs for each Example 1 token at the current target

Last tokenRequired value before itCore mode
Backward pruning: “Possible” here means only that the required input is a positive whole number; it does not by itself prove that the other tokens can reach that input. a token that would require zero or a negative previous value cannot be the last step in positive-whole-number mode.
3

Reconstruct Worked example 1

One printed route closes after six operations
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Inspect one complete solution. Predict each next value, then reveal it. You will arrange these same six tokens yourself in Mission 4.

original route ledger

Step0 / 6
Current value1
Final value—
Loop statuswaiting
Printed construction: 1 → 9 → 3 → 7 → 5 → 6 → 1, using +8, ÷3, +4, −2, +1, ÷6.
4

Build your own valid order for Worked example 1

Any order satisfying the stated checker mode is accepted
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Your six-token construction. Choose an order that returns to 1 with positive whole-number results throughout. You may use a different order from the worked route. The distinct-digit mode adds an optional rule.

About the optional count: the given 1 is already one one-digit visit. Add each one-digit value reached along the uninterrupted route, but exclude the final return to 1. Repeated visits count separately; the digit-tour mode also requires them to be different.

Tokens placed0 / 6
Positive-whole steps0 / 6
Optional: one-digit visits so far1 / 6
Final closurewaiting

Select a token, then choose a lettered slot (placing a used token swaps the two slots)

No token selected.
Place all six tokens.
5

Explain what a route proves

A worked route is evidence of existence. Testing every possible order is an optional stronger investigation.
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Reason it through

Place one of six tokens: five remain. To undo ÷3, multiply the result by 3: an output of 4 needs an input of 12. One successful route proves that at least one solution exists; it does not prove uniqueness.

Not run yet.
All operation orders—
Positive-whole closures—
One-digit closures—
Distinct digit tours—

Explore the valid orders

Run the audit to view solutions.
Notice: the worked example route is valid, but it is not the only positive-whole closure. The extra distinct-value requirement reduces the possibilities further.
6

Reconstruct original Practice 1

Nine operations travel through all nine one-digit values in the printed answer
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A new puzzle with nine tokens. The goal is still to return to 1. Trace this longer worked route before trying the same nine tokens yourself in Mission 7.

Reference-answer route

Step0 / 9
Current value1
Distinct digits visited1
Closed?waiting
Printed answer: 1 → 6 → 3 → 7 → 4 → 8 → 9 → 2 → 5 → 1, using +5, ÷2, +4, −3, ×2, +1, −7, +3, ÷5.
7

Solve Practice 1 and test stronger conventions

The core original-style closure is separate from the added digit-tour challenge
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Your nine-token construction. Start with ordinary positive whole-number closure. If you want an extra challenge, try keeping every visited value one-digit, or making all nine visited digits different.

About the visit counts: start with the given 1, then count values reached before the final return to 1. Repeated visits count separately in the one-digit count; the different-values count counts each whole number once. Only the uninterrupted route from the start is counted. These extra counts do not add a rule to ordinary closure.

Tokens placed0 / 9
Positive-whole steps0 / 9
Optional: one-digit visits so far1 / 9
Optional: different whole values so far1 / 9

Arrange all nine operation tokens; selecting a used token swaps its old slot with the chosen slot

No token selected.
Place all nine tokens.
Why the added modes matter. The worked example only asks for a closed chain and prints one route. Its route also happens to visit the digits 1–9 exactly once before returning to 1. The exact audit keeps those claims separate.
All 9! orders—
Positive-whole closures—
One-digit closures—
Distinct 1–9 tours—
Audit not run.
8

Transfer the strategy to a new five-token loop

Added practice: use backward pruning before trial and error
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A fresh five-token puzzle. These are different operations, so the earlier route cannot simply be copied. Find a new order that returns to 1; multi-digit intermediate values are allowed.
New puzzle: Start at 1 and arrange ÷5, +1, ÷4, ×6, +2 so that every intermediate result is a positive whole number and the loop returns to 1.

About this five-token puzzle

These totals describe all possible orders, regardless of your current entries.

Possible token orders120
Total possible complete solutions1

Your ring now

Tokens placed0 / 5
Final closurewaiting

Build the unique positive-whole closure

No token selected.
Work backward: the final ÷5 requires a 5 immediately before it.
9

Circular-chain workshop

Eight objective checks · all eight correct complete the mission
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10

Exit ticket and certificate

Five correct responses complete Lesson 28.1
Not complete