2006年

# 摘要或動機

1.直觀觀察:發現在每一個循環中，當n 等差α時，Aα,β,n,k 則等差α+β、n- Aα,β,n,k 則等差β。

2.分類:將其分類為cα,n，使當中有規律可求。

3.循環觀察:發現每個循環的尾數n- Aα,β,n,k 都小於β。

4.循環尾數:設計公式求出每個循環節的尾數n、留下數Aα,β,n,k 及n-Aα,β,n,k

5.倒推:由與循環節中有等差的性質，則可以由循環節的尾數，推論出循環節中的任意一數。

Joseph Sequence is the problem that discussed the situation of eliminating1 and
retaining1 in the circle formed by n people. Joseph Sequence has appeared a number
of times in National Elementary School and Middle School Science Fair in Taiwan
(as shown in the table below). Past national science fairs and researches on Joseph
Sequence

The publications，The Art of Programing，CONCRETE MATHEMATICS ，by the expert of mathematical
calculation in the IT industry，Donlad E. Knuth，has provided detailed explanation
on it. However, all of those only discussed eliminating 1 and retaining β or eliminating
α and retaining 1.

The researcher proposed “Problems unsolved by teachers” in the national competition,
and discussed the situation of eliminating α and retaining β in the circle formed
by n people. This study continued the summer project of 2005, and conducted research
on the question of when is the last kth person eliminated in a circle formed by
n people. In the paper, α, β, n and k were independent variables and the research
process was as follows:

1. Direct observation: the series shows equal difference in each cycle.

2. Classification: to search the pattern of the series based on cα,n classification.

3. Use the end number of each cycle to obtain the pattern.

4. Reverse induction: use the equal difference of each cycle to induce when the
kth person would be eliminated.