Formula and derivation of simple general solution for sailors ' coconut type problem (full version)

Transferred from: http://blog.sina.com.cn/s/blog_a1494e1301013w7v.html
Preface: "Sailor sub-coconut" is a fun math problem "sailors, monkeys and coconuts," the customary abbreviation, in China was changed (five monkeys peach) This is a world-famous interesting math problem, first published in the United States, the Saturday Evening Post, it is said that the first by the great physicist Dirac proposed, This seemingly simple problem has troubled him to get a simple calculation method, he gave the question to some of the mathematicians at that time, interestingly, unexpectedly also did not get satisfactory results, in 1979, the Nobel Prize Laureate in physics, Dr. T.D. Lee in the "China University of Science and Technology" lectures, specifically mentioned this topic; Since then, The simple calculation method of the problem is quickly swept into China. Once the "five sailors sub-coconut" widely circulated (mainly in foreign countries), played an important role, the famous modern mathematical logic theorists Whidehei, has used the theory of high-order difference equations and special solution of the relationship between the "sailor sub-coconut" A question, gave an answer for (-4) ingenious solution. In the last more than 10 years, with the continuous efforts of the latter, the method of solving the problem has been developed gradually. But strictly speaking: the results achieved at present are still local, only limited to the "Sailor sub-coconut" (or five-monkey peach) This specific topic, away from a comprehensive and simple solution to all this type of topic, there is a greater distance. I have in 1979 in the monthly "Chinese youth" to see the "Five Monkey Peach", and through the indefinite equation to find its solution, at that time, I think the topic of the meaning of the question is not very large. At the same time, in a very complicated calculation process, vaguely feel that this type of problem seems to be able to find a certain law. So through the five or six-day effort,   finally figured out all of this type of the complete, simple "general solution formula" (that is, all the factors affecting the answer can be arbitrary value) and can be very easy to solve. This has increased to a large extent (and can be said to be poor), to solve the "five Monkey Peach" type topic depth and breadth (see the calculation formula and examples below).            however, because at that time oneself in the country,  information Block,  also did not put this "general solution formula" is one thing. A more than 30 years later, the recent period of time, because of more idle, often on the internet,  so surprised to find: The search for "Five monkeys Peach" type of simple calculation method, is a deeper background, has been discussed for 20 or 30 years hot topic, and has not yet found the perfect solution. So his own side of the calculation, and finally re-deduced the "Five Monkey Peach" type of "general solution formula", and now publish it as follows, share with you:           complete and simple general solution formula for "sailor's Coconut" type problem :

y=an-db/c

y--The total number of things being divided,

a-the total number of points per minute (in general, the total number of people ),

N-The total number of points,

C-The number of copies taken after a part,

b-the remainder of a part after each

d-The number of copies left after each of the part A is taken out of C,

Note: When the b÷c is not a natural number, there is no solution at this time, i.e. y no solution. (with proof later)

the derivation process is as follows:

The number of things that the last person can see is; ax+b (x is the number of each serving after the last division a)

So the number of things the previous person saw was: (xa+b) a/d+b= a2x/d+ba/d+b

The first person to see a number of things: (a2x/d+ab/d+b) a/d+b= a3x/d2+b (A/d) 2+ba/d+b

Again, the previous person saw a number of things: A4x/d3+b (A/D) 3+b (A/d) 2+ba/d+b

The first person to see a number of things: y= a5x/d4+ (A/D) 4+ (A/D) (A/D) (a) (A/D) (a) (a) +1]b,

y= [xa5+ (A4+DA3+D2A2+D3A+D4) b]/d4

Recursive formulas according to the equal number of examples and sorted out after:

Y={xan +[an-1 (d/a) n/(1-d/a)]b}/dn-1

Y={xan +an-1[1-(d/a) n]ba/c}/dn-1

Y={xan+[an-1-(AN-1DN)/an]ad/c}/dn-1

y=[xan+ (AN-DN) b/c]/dn-1

Y= (XAN+ANB/C-DNB/C)/dn-1

Y= (XAN+ANB/C)/dn-1-db/c

y=an(x+b/c)/dn-1-db/c

A[(A/d) ^ (n-1) in the above-described section, if there is a (A/d) has the convention number must not numerator, otherwise a and D original definition does not exist, also cannot solve the problem. Therefore, the above should be further written:

Y=an[(X+B/C)/dn-1]-db/c

From the above can be seen: if the b/c is not a natural number, then (X+B/C)/dn-1 is not an integer, so the formula of the general solution is also no solution at this time, if the b/c is the natural number, then (X+B/C)/dn-1 will be able to obtain the minimum natural number 1, or any multiple of 1. Usually at the time of calculation, the minimum natural number 1 is generally taken, then the calculation and derivation of the above equation can be written as following simple general solution formula:

y=an-db/c

now use the above formula to solve, I in the last month Blog 12, 15, 16th out of three of this type of topic         

Example one , in "Nine Monkeys Peach": a=9, n=10, B=8, d=7, c=2

According to the general solution formula has: y=9 10 times Square -8x7÷2=3486784373.

example Two, in "16 sailors split coconut"