Drug problems in Rats

We have many bottles of colorless liquid, one of which is poison and the others are distilled water. After the experiment, the mouse will die five minutes later, the mice that drink distilled water are all normal. There are five mice now. How many bottles of liquid can we use these five mice for detection in five minutes?

 

Analysis: For convenience of analysis, we take three mice as an example to export the general formula.

First, we give each bottle from 1 ~ N number, from 0 ~ K is numbered (Here K = 3-1 = 2)

Map k + 1 mouse to each digit of the binary number according to its serial number, as shown below:

Mouse No.: K K-1... 0

Binary: 0 0... 0

 

Next, we feed the I-th bottle of liquid to the mouse that corresponds to the binary I, where the median is 1, as shown below:

	1 (001)	2 (010)	3 (011)	4 (100)	5 (101)	6 (110)	7 (111)
Rat 0	001		011		101		111
Rat 1		010	011			110	111
Rat 2				100	101	110	111

 

If five minutes later, the mouse died, And the binary position 1 corresponding to the mouse serial number can be determined which bottle of liquid is toxic,

For example, if 0th and 2nd mice die, the binary position 1 corresponding to the serial number is 101, that is, 5, that is, 5th bottles of liquid are toxic, if none of them are dead, it indicates that the serial number bottle is non-toxic. Then the total amount of liquid that can be measured is 23-1 = 7 bottles.

 

Similarly, for five mice, 25-1 = 31 bottles can be detected, and for k, 2k-1 bottles can be detected.