Grade 5 · Chapter 12 · Lesson 4

State Cycles: Warm-Up, Repeat, and Jump Ahead

状态周期：预备段、循环与快速跳步

Some processes look irregular at first and then settle into a loop. We will record the entire state, locate the warm-up and repeating part, and use an offset to jump safely to a very distant step.

Same complete state + same rule → the entire future repeats.

10 interactive missionsThree complete investigationsState tables and simulatorsEverything needed is on this page.

Mission 1

A repeated value is not always a repeated state

A state is all the information needed to determine the next step. In a five-box process, that means the ordered list (A, B, C, D, E)—not just one box.

Not complete

State P

A6B2C5D4E3

The least box is B.

State Q

A6B8C5D7E4

The least box is E.

Both states have A = 6, but the next move is different. One matching box count does not prove a cycle.

Read the two states

Do both states have the same value in Box A?

Are the complete states equal?

What must repeat to guarantee the same future?

Warm-up positions and offsets: choose one convention

If the cycle starts at Step s, measure N − s moves from that state. Remainder zero means return to Step s. If instead you list the cycle as items 1 through p, remove the warm-up items and use the final block item for remainder zero. Both methods work; do not mix their starting points.

A repeat interval need not be the shortest period. Look for the first later return of the full state when identifying its smallest cycle.

Mission 2

Separate the warm-up from the repeating cycle

A process may have a nonrepeating warm-up before a state repeats. Once the cycle starts, subtract the cycle-start step before dividing by the period.

Not complete

Use whole-number steps and a positive whole-number period. The questions below use the stated cycle start 3 and period 5, independently of these controls. The offset formula applies only at or after the cycle starts.

Cycle begins at step

Period

Target step

47target − cycle start

9complete cycles

2steps into cycle

5equivalent step

50 − 3 = 47 = 9 × 5 + 2 → Step 50 matches Step 5

Offset rule: if a cycle begins at step s and has period p, then a target step N at or after s matches step s + ((N − s) mod p).

The complete state at Step 3 first returns at Step 8. What is the shortest period?

With cycle start 3 and period 5, Step 18 matches which step?

With the same cycle, Step 50 matches which step?

Mission 3

Generate the reporting sequence and find its cycle

Complete practice problem: 2,016 students stand in a line. Student 1 reports 1. If a student reports a one-digit number, the next reports that number plus 9. If a student reports a two-digit number, the next reports its units digit plus 6.

Not complete

Extra practice

Highlight student 15

Warm-up: Student 1 reports 1Cycle begins at Student 210 repeats at Students 2 and 15

Warm-up

This first value does not belong to the repeating block.

Repeating block

10, 6, 15, 11, 7, 16, 12, 8, 17, 13, 9, 18, 14

Because the next report depends only on the current report, returning to 10 forces the same future. The period is 13.

How many warm-up terms are there?

What is the period?

Which report begins the repeating block?

At which later student does that report first repeat?

Mission 4

Jump directly to Student 2,016

The repeating block begins at Student 2, so remove the one warm-up term before locating Student 2,016 inside the 13-term cycle.

Not complete

Cycle arithmetic

2,016 − 1 = 2,015

2,015 = 155 × 13 + 0

A zero remainder means the 13th and last item of the cycle.

Cycle wheel

13-term cycle
remainder 0 → item 13

Answer: the 13th item of the repeating block is 14, so Student 2,016 reports 14.

After removing the warm-up, how many positions remain?

How many complete 13-term cycles?

What is the remainder?

What does Student 2,016 report?

Mission 5

Simulate the five-box minimum process

Complete box-process example: Boxes A–E begin with 9, 5, 3, 2, and 1 balls. On each move, find the box with the fewest balls. Take one ball from each of the other four boxes and place all four into the least-filled box. Move 0 is the initial arrangement; the display for Move n shows the counts after n complete moves. Box sizes in the drawing do not represent capacity. The same move-number convention applies to the other box models.

Not complete

Worked example 3

Move 0

Initial stateTotal balls = 20State (9, 5, 3, 2, 1)

Conservation check: four boxes lose one ball each and the least-filled box gains four. The total number of balls stays 20.

Which box is chosen on Move 1?

How many balls are in Box E after Move 1?

What is the total number of balls after every move?

Mission 6

Find the repeated state and jump to Move 50

Record complete states until one row repeats. In the worked example table, Move 3 and Move 8 both equal (6, 2, 5, 4, 3).

Not complete

Move	A	B	C	D	E	State note

Test a pair of rows

First move

Second move

Choose a row

↔

Choose a row

Select two moves to compare their complete states.

Find the period

8 − 3 = 5

The complete state repeats every five moves from Move 3 onward.

Jump to Move 50

50 − 3 = 47 = 9 × 5 + 2

Move 50 ↔ Move 3 + 2 = Move 5

Move 5 has state (4, 5, 3, 2, 6).

Cycle-start move

Period

Move 50 matches which earlier move?

How many balls are in Box B after Move 50?

Mission 7

Guided practice: a cycle that begins immediately

Practice box process: Boxes A–D begin with 6, 4, 5, and 3 balls. Children take turns, each making exactly one move. Each child finds the least-filled box, takes one ball from each of the other three boxes, and places the three balls into the least-filled box. Find the number in Box B after the 34th child.

Not complete

Guided Practice 3

Move 0

Initial stateTotal balls = 18State (6, 4, 5, 3)

State cycle

Move 0: (6, 4, 5, 3)

Move 4: (6, 4, 5, 3)

The cycle begins immediately and has period 4.

Jump to Move 34

34 = 8 × 4 + 2

Move 34 matches Move 2:

(4, 6, 3, 5)

Period

Move 34 matches which move in the first cycle?

Balls in Box B after Move 34

Mission 8

Redistribute from the greatest box

A greatest-box process: Boxes A–E begin with 2, 4, 6, 8, and 10 balls. On each move, find the box containing the most balls, remove four balls, and give one ball to each other box. Find the number in Box A after Move 2,014.

Not complete

Worked example 6

Move 0

Initial stateTotal balls = 30State (2, 4, 6, 8, 10)

Detect the cycle

Move 2 = Move 7 = (4, 6, 8, 5, 7)

period = 7 − 2 = 5

The warm-up consists of Moves 0 and 1; the cycle starts at Move 2.

Jump to Move 2,014

2,014 − 2 = 2,012 = 402 × 5 + 2

Move 2,014 ↔ Move 2 + 2 = Move 4

Move 4: (6, 8, 5, 7, 4)

original indexing note: the book labels Move 2 as the first state of the cycle, so it writes (2,014 − 1) ÷ 5 and gets remainder 3. That third cycle state is also Move 4. Both index systems give A = 6.

Cycle-start move

Period

Move 2,014 matches which earlier move?

Balls in Box A after Move 2,014

Mission 9

Independent workshop

Try each new problem before opening a hint. Correct all 8 answers to complete this mission.

Not complete

1. A state cycle starts at Step 5, with period 4. Which step from 5 through 8 matches Step 101?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

101 − 5 = 24 × 4 + 0. Add the remainder to 5: Step 5.

2. A state cycle starts at Step 2, with period 7. Which step from 2 through 8 matches Step 64?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

64 − 2 = 8 × 7 + 6. Add the remainder to 2: Step 8.

3. A state cycle starts at Step 0, with period 6. Which step from 0 through 5 matches Step 90?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

90 − 0 = 15 × 6 + 0. Add the remainder to 0: Step 0.

4. A complete state at Step 4 first returns at Step 10. What is its cycle length?

Hint

Subtract the earlier step from its first return.

Worked explanation

10 − 4 = 6.

5. A sequence begins with one special term, then repeats 2, 5, 8. What is term 40?

Hint

Remove the one-term warm-up before using the three-item block.

Worked explanation

40 − 1 = 39 = 13 × 3. Use the final block item, 8.

6. Can comparing Step 5 with itself prove a cycle? Write yes or no.

Hint

A cycle must return at a later time.

Worked explanation

The difference is zero. A cycle needs equal states at two distinct steps.

7. A state cycle starts at Step 3, with period 5. Which step from 3 through 7 matches Step 43?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

43 − 3 = 8 × 5 + 0. Add the remainder to 3: Step 3.

8. The rule depends on boxes A, B, and C. Only A has repeated. Has a cycle been proved? Write yes or no.

Hint

Ask whether all information controlling the next move matches.

Worked explanation

B and C may differ. Compare the complete ordered state.

Mission 10

Transfer exit ticket

Try each new problem before opening a hint. Correct all 5 answers to complete this mission.

Not complete

1. A state cycle starts at Step 6, with period 5. Which step from 6 through 10 matches Step 103?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

103 − 6 = 19 × 5 + 2. Add the remainder to 6: Step 8.

2. A state cycle starts at Step 0, with period 4. Which step from 0 through 3 matches Step 76?

Hint

Subtract the cycle-start step before taking a remainder.

Worked explanation

76 − 0 = 19 × 4 + 0. Add the remainder to 0: Step 0.

3. After two special starting terms, a sequence repeats 4, 7, 9, 1. What is term 30?

Hint

Remove the two starting terms, then use the four-item block.

Worked explanation

30 − 2 = 28 = 7 × 4. The last item is 1.

4. A state at Step 8 first returns at Step 14. Find the period.

Hint

Count the moves between these matching states.

Worked explanation

14 − 8 = 6.

5. Two states have the same total but different box counts. Must their futures agree? Write yes or no.

Hint

Does the total tell you which box the rule selects?

Worked explanation

No. The full ordered counts determine the next move, not just their total.

Optional reflection — not automatically graded

Why is matching the complete state stronger than seeing one number repeat?

Lesson checkpoints completed

State-Cycle Time Jumper

This certifies that the learner can identify a complete state, separate a warm-up from a cycle, locate a repeated state, and jump safely to a distant step using an offset and remainder.

Lesson 12.4 · Grade 5 Periodic Problems

Teaching notes

The greatest-box redistribution problem, including the repeated states at Moves 2 and 7 and the value in Box A after Move 2,014, comes from Example 6 across pages 78–79.This page restates every rule and numerical condition in English and adds interactive state models, an explicit cycle-offset formula, objective practice, and feedback.