Combine Cycles and Track Moving Positions
Lesson 12.6 — 组合周期与循环位置
Follow toys, symbols, and moving markers through repeating systems. Learn when to use a remainder, when to use a least common multiple, and when to track each cycle separately.
Build the first shared restart
A four-step cycle restarts at 4,8,12,16,… . A six-step cycle restarts at 6,12,18,… . Their first common restart is 12. This is the least common multiple, written LCM(4,6) = 12. Adding the lengths gives 10, which is not a restart of either cycle.
Read a signed remainder in ordinary language
“19 mod 8 = 3” means “19 leaves remainder 3 when divided by 8.” On an eight-position ring, five steps counterclockwise reach the same position as three steps clockwise. That is why −5 mod 8 is written as 3 in the movement formula.
What must repeat in an alternating schedule?
The state includes the location and which move comes next. A return to a location on a different odd/even day may have a different future. In the worked example four-day return, the marker is back at 1 and the next move is again the odd-day clockwise move, so the full state repeats.
Track one moving position with a signed remainder
Positions are numbered clockwise, starting at 1. One step goes to the next marked position; the starting position is not counted as a step. A circular path repeats after one complete lap. Clockwise and counterclockwise moves must be counted in opposite directions.
Start at 1 and move 19 steps clockwise.
Use a zero-based position inside the calculation
On an 8-position circle:
Two complete laps change nothing. The remaining three steps move 1 → 2 → 3 → 4.
Circle-position laboratory
S marks the start; L marks the landing. Use the fixed 19-step example for the questions below.
Check the 19-step example
Break the toy-passing rule into smaller cycles
Six children pass their current toys according to one fixed rule every round. The arrows themselves reveal two smaller cycles.
The complete original rule
Children A, B, C, D, E, and F begin with their own toys. Round 0 is the starting arrangement. In each round everyone passes simultaneously, once: the toy received is kept until the next round. Each child sends the toy currently in hand as follows:
The toy movement separates into these cycles:
Cycle lengths
The A–F–C–E cycle has length 4. The B–D cycle has length 2.
Why use the least common multiple?
The whole arrangement is back only when both smaller cycles are back at the same time. Four rounds works for both lengths.
Record the cycle information
Watch the complete toy arrangement repeat
A repeated toy at one child is not enough. We compare the ordered state of all six holders.
| Round | A holds | B holds | C holds | D holds | E holds | F holds |
|---|
Read Round 1 and Round 4
Jump to Round 2,014
Once the full state has period 4, a very large round number becomes a small remainder problem.
Compress 2,014 rounds
Round 2,014 has the same arrangement as Round 2.
Only B and D are holding their own toys.
Any target round
Complete the worked example result
Combine two repeating rows with the least common multiple
The worked example uses two repeating Chinese phrases. No Chinese reading is needed: each character can be treated as a symbol with a numbered position.
original-style paired pattern
A group is one top symbol paired with the bottom symbol directly beneath it. Group 1 uses the first symbol of each row, and both rows advance one place per group. The top row has 8 symbols and the bottom row has 9. Their full pair pattern restarts after:
Top row — eight characters from a phrase meaning roughly “Mathematics is exercise for the mind.”
Bottom row — nine characters from a phrase meaning “We take part in the Hope Cup contest.”
Analyze Group 2,015
Locate paired cycles separately when that is faster
A practice problem pairs A, B, C, D with 1, 2, 3. You may use the 12-group combined cycle, or reduce 26 by 4 and by 3 separately.
The two rows
The paired pattern repeats after:
Group 26
Paired-cycle laboratory
Both rows begin with their first item at Group 1 and advance together. The strip previews at most the first 24 groups of a cycle. Answer the fixed A–D / 1–3 questions below using Group 26.
Complete the practice problem
Track an alternating movement schedule
Positions 1–8 are numbered clockwise. Before Day 1, a marker starts at Position 1 on the circle. Odd days move 329 positions clockwise; even days move 485 positions counterclockwise.
Reduce each move before tracking
So the positions are:
The marker first returns to Position 1 after 4 days.
| Day | Move used | Effective clockwise move | Position |
|---|
Analyze the four-day return
Track two movers independently on the same ring
Two jumping insects start at the position labeled 12 on a 12-position clock ring. They move in opposite directions by different numbers of steps.
Red: 1,991 clockwise steps. Black: 1,949 counterclockwise steps.
Reduce each jump count separately
General circular-movement laboratory
Positions increase clockwise. S marks the start and L the landing. The starting position must be between 1 and the number of positions. The questions below use the stated insect movements.
Complete the worked example landing calculation
Independent workshop
Try each new problem before opening a hint. Correct all 8 answers to complete this mission.
Hint
Remove whole laps, then follow the stated direction. The start is not a move.
Worked explanation
32 = 3 × 9 + 5. From 4, move 5 positions clockwise to 9.
Hint
Remove whole laps, then follow the stated direction. The start is not a move.
Worked explanation
27 = 2 × 10 + 7. From 3, move 7 positions counterclockwise to 6.
Hint
List positive multiples of both lengths.
Worked explanation
6,12,18 and 9,18 first meet at 18.
Hint
Find the arrow ending at C.
Worked explanation
F passes to C, so C holds toy F.
Hint
Reduce 38 separately by five and three.
Worked explanation
38 mod 5 = 3 and 38 mod 3 = 2: C2.
Hint
Remove whole laps, then follow the stated direction. The start is not a move.
Worked explanation
11 = 1 × 8 + 3. From 1, move 3 positions clockwise to 4.
Hint
Group the first four days, then perform the next odd-day move.
Worked explanation
Four days move 1+3+1+3 = 8, returning to 1. One more step lands at 2.
Hint
Use the first common multiple, not the sum.
Worked explanation
3,6,9,12 and 4,8,12 first meet at 12.
Transfer exit ticket
Try each new problem before opening a hint. Correct all 5 answers to complete this mission.
Hint
Remove whole laps, then follow the stated direction. The start is not a move.
Worked explanation
29 = 2 × 12 + 5. From 5, move 5 positions counterclockwise to 12.
Hint
List multiples or use the product for these coprime lengths.
Worked explanation
The first common multiple of 5 and 6 is 30.
Hint
Reduce the index separately by three and four.
Worked explanation
35 mod 3 = 2 and 35 mod 4 = 3: B3.
Hint
Apply the arrows three times, or read the holder table.
Worked explanation
Toy F travels F → C → E → A. After three rounds A holds F.
Hint
Remove whole laps, then follow the stated direction. The start is not a move.
Worked explanation
20 = 2 × 7 + 6. From 2, move 6 positions clockwise to 1.
Optional reflection — not automatically graded
Cycle-Combination & Circular-Movement Navigator
All ten lesson missions completed: track full states, combine cycles, and locate moving positions.
Teaching notes
The toy-passing investigation follows Chapter 12, Example 4.The two repeating phrase rows follow Guided Practice 5.The A-D/1-3 pair, alternating 8-position movement, and two-insect clock-ring problems follow Exercises 9–11.The general laboratories, English indexing labels, feedback, workshop, and exit ticket are new instructional scaffolds.