Puzzle | Finding the Injection for Anesthesia

In a Medical Laboratory:

• You have 240 injections, and exactly one of them is an anesthetic.
• You also have 5 rats to help identify which injection is the anesthetic.

How it works

• If a rat receives the anesthetic injection, it will faint sometime within 24 hours after that injection (the exact time is unknown; could be any time ≤ 24 hours).
• Once a rat faints, it cannot be used again.
• Multiple injections can be given to a rat at different times until it faints.

How can you design a testing strategy to find the anesthetic injection within 48 hours using only 5 rats?

Rules:

• You have a total of 48 hours to identify the single anesthetic injection.
• You may observe and record fainting times during the 48-hour period.
• You must guarantee identification of the anesthetic within the 48 hours (i.e., the scheme must work in the worst case).

Check if you were right - full answer with solution below.

Solution: 
Let us number each injection with unique 5-digit numbers consisting of only 0, 1, and 2.
Let us number the Rats as 1, 10, 100, 1000, 10000. 

• The action corresponding to the digit in the unit place will be executed by rat numbered 1.
• The action corresponding to the digit in the tenth place will be executed by rat numbered 10.
• The action corresponding to the digit in the 100th place will be executed by rat numbered 100.
• The action corresponding to the digit in the 1000th place will be executed by rat numbered 1000.
• The action corresponding to the digit in the 10000th place will be executed by rat numbered 10000.

Number meanings on an injection:

• Number 0 on an Injection represents that the injection will not be injected to the rat. 
• Number 1 on an Injection represents that the injection will be injected to the rat on the 1st day. 
• Number 2 on an Injection represents that the injection will be injected to the rat on the 2nd day (after 24 hours). 

Example:

Let us say the Injection is numbered 11201. The Injections is injected on the first day to rat numbered 10000, 1000, and 1. It is injected on the second day to rat numbered 100. And it is not injected to rat numbered 10. 

So if the rats numbered 10000, 1000 and 1 faint within first 24 hours, rat numbered 100 faints in the next 24 hours and the rat numbered 10 does not faint, then the Anesthesia Injection has to be 11201. 

Total number of injections that can be tested in this way will be:

= 3 * 3 * 3 * 3 * 3 = 3^5 = 243 Injections

So with the help of 5 rats and within 48 hours we will be able to find a Anesthesia Injection among 243 Injections.