The formula for the general solution of five monkeys--presented to Nobel laureate Dr. T.D. Lee

Abstract: "Five monkeys divide Peach problem" is a very famous in and out of the interesting math problem. The simple method of calculating this type of problem has plagued some large physicists and mathematicians. Dr. T.D. Lee a lecture at the Chinese University of Science and Technology, and also specifically mentioned this question,

Through the analysis of this problem, this paper deduces the simplest general solution Formula y=a (a/m) n-1-db/c, and its sister formula: Y=[ka (a/m) n-1-db]/c, which can solve all of these types of problems. And the two formulas have the solution and no solution, so that the problem has been more than the full settlement.

Summary: "Five Monkey Peach Assignment Problem", formerly known as "Sailor of the Coconut," It is a well-known Chinese and Foreign fun math puzzle. Studies find its simple calculation method has plagued a number of physicists and mathematicians. Dr. Li Zhengdao lectures at the Chinese, but also specifically mentioned the problem.
The author, through the analysis of the problem, got to solve this kind of problem solving systems and formulas to make th Is mathematical problem have been half a century, to get a better solutio

Preface: "Five Monkeys divide Peach problem" is the predecessor of the famous foreign "sailor sub-coconut Problem", the drama said, the first by the great physicist Dirac in 1926, and then, after the American Mathematical Science master Martin Gardner Introduction, promotion, the problem has been more widely circulated. In 1979, "Nobel Prize" winner Dr. T.D. Lee, in the "China University of Science and Technology" lectures, specifically mentioned this topic. Since then, the study of the simple calculation method of the problem, quickly swept the domestic

Once the "sailor sub-coconut" has played an important role in the widespread, the famous modern mathematical logic biologist Whidehei, to this question gave an answer for (-4) ingenious special solution. In the continuous efforts of the latter, some relatively simple methods are gradually appearing. But strictly speaking: the results achieved at present are basically confined to the specific topic of "Five Monkeys and Peach", and there is still a great distance to solve all the problems of this type from the comprehensive and simple.

In 1979, I was fortunate to see a "five-monkey-peach" in the Monthly "Chinese Youth", and to obtain its solution by indefinite equation. Then the calculus is deduced to solve the core problem solving formula of this kind of question:y=an-db/c . But not until the previous period of shock AH found that: the search for "Five monkeys Peach" type of simple calculation method, is a deeper background, domestic and foreign has been a hot topic for decades, and still have not found a better solution. Therefore, I have further perfected the solution system of the simple general solution formula by continuing the analysis of this problem, and now we share it with you:

I. Five simple general solution formula for the type problem of monkey

General Solution Formula (1), Y=a (a/m) n-1-db/c, (when B/C is a positive integer)

General Solution Formula (2), Y=[ka (a/m) n-1-db]/c, (when B/C is not a positive integer)

which

y── total number of peaches to be divided

n── total number of points

a── the number of copies per minute,

b── the remainder of a part after each.

c── the number of copies to be taken after each sub-part A,

d── take c after each sub-part of a, and then the number of copies remaining.

m--(A/D) is a greatest common divisor including m=1

A parameter in the k--formula (2) that enables the Y value to be an integer

Note:

(a) in the test instructions formula, the requirements for the type of question are as follows: Y, A, B, C, D, N, M, are all positive integers (otherwise meaningless)

(b) for the formula (1), if the b/c is a positive integer, then the general solution must have a solution. If the b/c is not a positive integer, it is solved with the general solution formula (2),
(c) for the formula (2), if the b/m is a positive integer, then the general solution must have a solution. If b/m is not a positive integer, then the general solution formula (2) has no solution. ,
(d) The general Solution formula (2) in the K, can be obtained by the K formula: K= (fc+b) (m/d)n-1 , (test k is greater than or equal to 0, F is the natural number that can make K rounding, under normal circumstances, K will be less than =c), and then the K value into the formula 2, It is easy to get the solution directly.

In the above solution formula, if we take m as 1, take db/c as an integer, then it shows the form is the original core formula:y=an-db/c

Derivation and argumentation of the formula of general solution

1 , derivation of formula example "nine Monkey Peach"

As the "Five Monkeys in peach" in my derivation of the "general solution formula", has become a very simple calculation problem, not enough to explain the formula for this type of problem, all-round general solution ability, so increase the difficulty and complexity of the topic, changed to "Nine Monkey Peach", the title is as follows:

One day there were 9 monkeys busy all day, picking a bunch of big peaches, are too tired to fall asleep. At night, a monkey quietly get up, the peach divided into 9 parts, the results found that more than 8 peaches and then it ate the 8 peaches, and greedy to take away 9 of the 2 parts, and then put the remaining peaches mixed together put back in place, quietly back to bed.

After a while, the other monkey also quietly get up, the remaining peaches are divided into 9 parts, the result is just extra 8 peaches, it also ate the 8 peaches, and then hid 9 of the 2 parts, the remaining peaches mixed together, also quietly drip back to sleep.

After a while ...

After a while ...

When the 7th monkey is just like the 6 monkeys in front, the peach is divided into 9 parts, proud to eat the extra 8 peaches, then suddenly a few tigers roar, frighten 9 monkeys even jumping, fled. Now, please, how many of these peaches do you have at least?

2 , the derivation and demonstration of general solution Formula
Derivation method One

Set: The number of peaches divided into a total of y , the total number of points per cent is a, the remainder is B. After each division a is taken to C , the rest of the number of copies of B , the total number of times is N times, the last person divided a Each copy of the copy is x (positive integer x)

Well, the last one to get a peach to the monkey, see the number of Peaches is: ax+b

The number of peaches seen by the last Monkey is: (xa+b) a/d+b=a2x/d+ba/d+b.

The number of peaches seen in the last monkey is:

(a2x/d+ab/d+b) A/d+b=a3x/d2+b (A/D) 2+b (A/D) +b.

Similarly: The last monkey saw the number of Peaches: A4x/d3+b (A/D) 3+b (A/D) 2+b (A/D) +b.

Also, the first monkeys saw the number of peaches:

A7x/d6+[(A/D) 6+ (A/D) 5+ (A/D) 4+ (A/D) (A/D) (A/a) (/a) +1]b.

The formulas are recursive and sorted according to the equal number examples:

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

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

=[anx+ (An-1-an-1dn/an) ad/c]/dn-1

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

= (ANX+ANB/C-DNB/C)/dn-1

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

the basic equation of the solution is obtained. Y=an (x+b/c)/dn-1-db/c, and from this can be obtained, a simple formula and two general solution formulas, respectively, are described as follows:

(1) when the type of a (A/d) n-1 part, if (A/D) No convention number, then an and dn-1 coprime, so the above can be further written: y=an[(x+b/c)/dn-1]-db/c

From the above can be seen: according to test instructions dn-1 is necessarily a positive integer, when (B/C) is also a positive integer, then (X+B/C)/dn-1 will be able to obtain the minimum natural number 1, or 1 of any integral multiples, that is y=kan-db/c (b /c) is a positive integer formula must have a solution, usually in order to simplify the calculation, K generally take the minimum natural number 1, then the above equation can be simply written, simple formula:y=an-db/c, This formula can be regarded as a general solution of this type of problem, But not necessarily the smallest solution.

(2) if the occurrence (A/d) has the Convention number this situation, at this time the y value, but also has the formula,y=an-db/c Smaller solution,

Now we continue to Y=an (X+B/C)/dn-1-db/c, this step continues to verify, set M for (A/b) greatest common divisor, there are:

y=a[(a/m)/(d/m)]n-1 (x+b/c)-db/c

=a (a/m) n-1 (x+b/c)/(d/m) n-1-db/c.

According to the same reasoning behind the first of the above, you can get: Y=a (a/m) n-1-db/c

Obviously, if we think of 1 as the number of conventions (A/D), then Y=a (a/m) n-1-db/c=an-db/c when the number of conventions (A/D) is only 1 o'clock.

In other words: the latter is essentially the former special form, and Y=a (a/m) n-1-db/c, is when the b/c is an integer, the solution of all the minimum solution of this type of the general solution formula. Its solution set is: Y=ka (a/m) n-1-db/c

(3), if the b/c is not a positive integer, then the general solution formula can be used, Y=[ka (a/m) n-1-db]/c. To solve, the formula is deduced as follows:

Method one , for the underlying equation: Y=ka (a/m) n-1 (x+b/c)/(d/m) n-1-db/c. , the numerator and denominator of the "A (a/m) n-1 (x+b/c)/(d/m) n-1" portion of which can be left with C, the Y=ka (a/m) n-1 (X+B/C) c/d (d/m) n-1-db/c, H, and (X+B/C) c=k (d/m) N-1 (k is a positive integer) such h, it becomes Y=ka (a/m) n-1/c-db/c, and further obtained; Y=[ka (a/m) n-1-db]/c. (i.e. general Solution Formula 2)

Method Two deduces the previous line of the two basic equation according to the formula: Y=can (X+B/C)/cdn-1-db/c, if we make (and must have) C (x+b/c) =kdn-1 (k is a positive integer that can make C (x+b/c)/dn-1 a positive integer), you can also get y= [KAN-DB]/C, such as a and d have the convention number, can also be further written general solution Formula Two: Y=[ka (a/m) n-1-db]/c.

(4) about Equation 2, the derivation of K formula; Set X=[k (d/m) n-1-b]/c, there is cx+b=k (d/m) n-1 so there is k=cx+b/(d/m) n-1, the last obtained: K= (fc+b) (m/d) n-1 (k is a positive integer, F is the natural number that can make the K integer)

With this formula of general solution, A (a/m) n-1-db/c and the analysis of the key points of the derivation process, we can easily get it. General Solution Formula derivation method two and derivation method three.

Derivation Method Two

Set: When the first allocation, take the quantity is: (Y-b) c/a;

The remaining AAUs are: (y-b) d/A.

The second allocation when the quantity is taken; {[(y-b) c/a]-b}c/a;

Then the remaining number of redistribution is: [(y-b) d/a-b]d/a= (y-b) (D/A2)-bd/a.

Also: The third time, the remaining number of redistribution is: (y-b) (d/a/) 3-b (d/a) 2-bd/a,

The fourth time, the remaining number of redistribution is: (y-b) (d/a/) 4-b (d/a) 3-b (d/a) 2-bd/a,

Then push backwards ... The last available: the number of the remaining redistribution for the seventh time is:

x=[(Y-b) (d/a) 6-b (d/a) 5-b (d/a) 4-b (d/a) 3-b (d/a) 2-b (d/a)-b]/a

yd6/a7-x=[(d/a) 6+ (d/a) 5+ (d/a) 4+ (d/a) (d/a) (d/a) +1]b/a

According to the series of recursive formula and organized after:

X=ydn-1/an-{1[1-(d/a) n]/(1-d/a)}b/a

X=ydn-1/an-[1-(d/a) n]b/c

X=ydn-1/an-b/c-bdn/can

X=dn-1/an (Y-BD/C)-d/c

The same basic equation is obtained: Y=an (x+b/c)/dn-1-db/c,

This derivation (ii) although compared to the derivation (a) to be simple, but the last two steps have a great jump, and do not conform to the normal thinking of deriving such questions, it is difficult to think. Just because we have seen that the current formula shape is very close to the target formula y=a (a/m)n-1-db/c shape , We will think that can be deduced.

The following formula calculus, in the same way as the derivation method, is then y=A (a/m)n-1(x+b/c)/(d/m)n-1-db/c. This part continues to be calculated, and finally can be obtained simple general solution formula:y=a (a/m) n-1-db/c and Y=[ka (a/m) n-1-db]/c.

Derivation Method Three

Set: The 7th time the Peach Monkeys see the number of Peaches: Ax+b, (X is, the last time a portion of each part of the number).

The 6th time the Peach Monkeys see the number of Peaches: (ax+b) a/d+b=a2x/d+ba/d+b.

Also, the 5th time the Peach Monkeys see the number of peaches:

(a2x/d+ab/d+b) A/d+b=a3x/d2+b (A/D) 2+ba/d+b.

Then, all the way back to push, you can get the first point of the peach monkeys to see the number of peaches:

Y=a7x/d6+[(A/D) 6+ (A/D) 5+ (A/D) 4+ (A/D) (A/D)

Within the brackets is a common ratio (A/d) of the equal number of examples, according to the equal number of the recursive formula is:

y=anx/dn-1+{(A/d)n-1[n (d/a) n/(1-d/a)]}/b

=anx/dn-1+[(A/d)n-1-d/a]ab/c

=anx/dn-1+ban/cdn-1-db/c

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

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

Here is the same as the derivation method one, can also get the general solution Formula One:y=a (a/m) n-1-db/c and general solution Formula Two,y=[a (a/m) n-1-db]/c

Third, PostScript

In this paper, to solve the problem of "five monkey Peach Problem" Simple calculation method, out of a single, local thinking of the problem, from the point of the law to find the overall solution, get a simple general formula (1):y=a (a/m)n-1 -db/c. and its sister formula (2):Y=[ka (a/m)n-1-db]/c. in the general solution formula,

1, because A and n can be any number, and other factors affecting the results of the calculation can be variables; So this formula is very poor and solves the depth and breadth of such problems.

2, each variable basically does not have the correlation coefficient computation, and basically all is appears separately, moreover the formula has the solution the condition which is also simple and clear, therefore this general solution formula should be solves this type question the simplest formula.

3 The solution obtained by the formula of general solution is the requirement of solving the minimum solution of this kind of problem, so it can be said that the mathematical problem which has been proposed for more than half a century has been solved more than the garden full.

Original address: http://blog.sina.com.cn/s/blog_a1494e130101eljh.html

The formula for the general solution of five monkeys--presented to Nobel laureate Dr. T.D. Lee