Quality exception ball problems in 12 small balls measured three times without Weight Balance

Quality exception ball problems in 12 small balls measured three times without Weight Balance

Original Title:
There are 12 small balls with the same features, and only one of them has a quality exception. It is required to be called three times without a weight to find the ball with an abnormal quality.

Solution:
Set the standard ball quality to W, and represent any normal ball. Number the 12 balls A1, A2,..., and A12 in sequence:
A1, A2, A3, and A4 are A1 groups.
A5, A6, A7, and A8 are A2 groups.
A9, A10, A11, and A12 are A3 groups.

= (First time) 1 select any two groups -- Take a1, a2 for comparison, if
1 A1 = a2 the A3 group is an exception ball group.
Regroup:
B1: A9 A10
B2: A11 W
B3: A12 W

==== (The second time) take any one group of B2 B3 -- B2 and B1 For comparison. If
1.1 b1 = b2 B1 B2 is the normal group, B3 (A12, W) is the exception group, and the exception ball is A12
1.2 B1! = B2 B3 (A12, W) is the normal group. The following uses B1 <B2 as an example:
Expression exp0: A9 + A10 <A11 + W

======== (The third time) Take A9 A10 for comparison. If
1.2.1 if A9 = A10, A11 is the exception ball.
1.2.2 A9! = A10 indicates that A11 is the normal ball. According to exp0, A9 + A10 is obtained <2 W
Therefore, the quality of the abnormal ball is smaller than that of the normal ball.

2 A1! = A2, then A3 (A9, A10, A11, A12) is the normal group; A1 Expression 1: exp1: a1 + A2 + A3 + A4 regrouping:
B1: A1, A2, A3
B2: A4, a5, A6
B3: A7, A8, W

== (second time) comparison between B3 and b2
2.1 B3 = b2
A4 = A5 = A6 = A7 = A8 = W based on exp1
A1 + A2 + A3 <3 W abnormal ball quality less than standard ball

======== (third time) compare A1 A2. If
2.1.1 a1 = A2, A3 is the exception ball.
2.1.2 A1! = A2, A3 is the normal ball, and A1 and A2 are the normal ball.

2.2 B3> B2, B1 is in the normal group
A1 = a2 = a3 = W get according to exp1
3 W + A4 <A5 + A6 + A7 + A8
(B3 <B2) A4 + A5 + A6 <A7 + A8 + W
Add 3 W + 2a4 + A5 + A6 <A5 + A6 + 2a7 + 2a8 + W
2a4 <A7 + A8
========= (Third time) Compare the A7 A8
2.2.1 A7 = A8 A4 is an abnormal ball with a quality smaller than the standard ball
2.2.2 A7! = A8 A4 is a normal ball. We can see that 2 W <A7 + A8, and the major in A7 A8 is an abnormal ball.

2.3 b2 <B3 then B1 is in the normal group
A1 = a2 = a3 = W get according to exp1
3 W + A4 <A5 + A6 + A7 + A8
(B2 <B3) A4 + A5 + A6> A7 + A8 + W
Conversion:-A4-A5-A6 <-A7-A8-W
Add 3 W-A5-A6 <A5 + A6-W
A5 + A6> 2 W
We can see that the quality of the abnormal ball is higher than the standard ball.
========= (The third time) compare with A5 A6
2.3.1 A5 A6 indicates an abnormal ball.

The preceding sections show that A1, A2,..., and A12 indicate that the probability of an exception ball is 1/12.

 

Trackback: http://tb.blog.csdn.net/TrackBack.aspx? Postid = 414490