Track Repeated Elimination with Remainder Classes
Lesson 15.5 · 用余数类追踪重复淘汰
A circle with thousands of pieces looks impossible to simulate. Number the positions, preserve the counting phase, and turn repeated “skip one, remove one” moves into simple remainder patterns.
Turn the moving rule into numbered positions
The pieces move nowhere; only our attention moves. Numbering the positions turns a physical process into arithmetic.
The exact rule
Begin at the black piece. Move clockwise. Skip the next living piece, remove the following living piece, and continue without restarting the count after a full circuit.
Removed pieces no longer count when skipping or removing. Keep the original labels; never renumber survivors. Stop immediately when black is removed. One removal is one operation; skips do not count as removal operations. If black is the only piece left, continue the same rule: skip it once, then remove it.
Number a 13-piece model
Place 12 white pieces after the black piece, moving clockwise. Number the white pieces:
Give the black piece the next label:
This small model has the same structure as the practice problem, but every position can be seen.
Check the setup
Watch the complete 13-piece process
A small simulation reveals the phase change that is easy to miss at the end of the first circuit.
Read the completed model
Apply the first circuit to 2,013 positions
Now use the complete practice problem. The ring contains one black Go piece and 2,012 white Go pieces.
One black piece and 2,012 white pieces
Number the white pieces from 1 through 2,012 clockwise, beginning with the first white piece after black. Number the black piece 2,013. Starting from black, skip one living piece and remove the next. How many white pieces remain when black is removed?
First circuit: remove the evens
These are precisely the positions with:
The count is:
What remains after the first circuit?
There are 1,006 odd-numbered white pieces plus the black piece:
Reveal the first-circuit ledger
Check the first circuit
Keep the counting phase through the second circuit
The key move occurs after white piece 2,012 is removed: black is the next living piece, so black is skipped and white piece 1 is removed.
Second-circuit removals
They differ by 4 and all satisfy:
The black piece is included because:
White positions still alive
They differ by 4 and all satisfy:
Every even position was removed earlier.
Four remainder classes modulo 4
0 mod 4
4, 8, 12, …
1 mod 4
1, 5, 9, …
2 mod 4
2, 6, 10, …
3 mod 4
3, 7, 11, …
Check the second circuit
Count the surviving arithmetic sequence
Knowing the survivor class is not enough. Count every term from the first survivor through the last, including both endpoints.
Direct count
Second ledger check
After the first circuit, 1,007 pieces are alive. The second-circuit removal sequence:
contains:
pieces, including black. Thus:
Arithmetic-sequence counter
Enter nonnegative whole-number first and last terms with last ≥ first, and a positive whole-number step. The last term must be reached exactly. For this mission, run the counter with 3, 2011, and step 4.
Complete both counts
Read a position from its remainder class
The notation n ≡ r (mod 4) means that n leaves remainder r when divided by 4.
Preview of original-ring positions 1 through 24 (not a separate 24-piece ring)
3 mod 4
Positions such as 3, 7, 11, 15, 19, 23, … survive until black is removed.
1 mod 4
Positions such as 1, 5, 9, 13, 17, 21, … are removed during the second circuit.
0 or 2 mod 4
These are the even positions, removed during the first circuit.
Classify these positions in the worked example ring of 2,013 pieces
Build a shortcut for 4m white pieces
The worked example numbers are not accidental: 2,012 = 4 × 503. The same two-circuit structure works for W=4m white pieces, where m is a positive whole number. Every survivor count in this lesson is measured at the instant black is removed, not after eliminating every piece.
Why the formula works
If there are 4m white pieces, the black label is 4m+1.
- First circuit removes 2m even positions.
- Second circuit removes m+1 positions of the form 4k+1, including black.
- The remaining white positions are 3, 7, …, 4m−1: exactly m pieces.
original as one family member
Use the family rule
Use a simulator when the shortcut does not fit
The W÷4 shortcut is proved only when W is a multiple of 4. For other ring sizes, simulate the same skip/remove rule and keep the phase exact.
General ring simulator
Run the simulator to display the ring at the instant black is removed.
Run both the 10-white and 6-white simulations, then compare
Remainder-class workshop
Answer all eight questions. Correct all eight responses to complete the workshop.
Worked explanation for question 1
White labels are 1–28, so black is 29.
Worked explanation for question 2
Remove the 14 even labels 2 through 28.
Worked explanation for question 3
28=4×7, so the family rule gives 7.
Worked explanation for question 4
The family has m+1=8 second removals.
Worked explanation for question 5
(59−3)÷4+1=15.
Worked explanation for question 6
87=4×21+3; it belongs to the survivor class in the worked example ring.
Worked explanation for question 7
Skip black 29, then remove white 1.
Worked explanation for question 8
124÷4=31.
Objective exit ticket
Complete all ten missions, including a perfect exit score, to earn the certificate.
Worked explanation for question 1
36=4×9.
Worked explanation for question 2
m+1=9+1=10.
Worked explanation for question 3
(99−3)÷4+1=25.
Worked explanation for question 4
101=4×25+1; the black label of a 100-white ring is removed second circuit.
Worked explanation for question 5
236 is divisible by 4, so 236÷4=59.
Certificate of mathematical thinking
Remainder-Class Elimination Navigator
has completed Lesson 15.5 by translating a circular elimination process into numbered positions, residue classes, and exact survivor counts.
Why must the count continue across the starting point?
Explain what would go wrong if a student restarted “skip one, remove one” after each circuit instead of preserving the phase.
About this lesson
The 13-piece simulator, the explicit remainder-class diagrams, the 4m family proof, the arbitrary-size simulator, workshop, and exit ticket are added instructional scaffolds. They preserve the worked example rule and make the counting phase visible.