This pandemic has made us realize the significance of finding a cure as soon as the disease breaks out. Any medicine that is designed to tackle a disease, takes some time to show its effects on the body. This, if not done properly can have severe side effects, in turn further delaying the cure. The sooner we get to a solution, more the number of deaths that we could prevent. This puzzle is one such thing.
We know how lab rats/mice and humans are inseparable. The majority of times, a clinical trial on the human body will not be done before the same has been tested on a rat. What if you are given 5 rats and a drug to use on them. Well, we don’t want to kill these little friends of ours. So instead of some random medicine with unknown side effects, let us use anaesthesia(for rats) as the drug here. You will be given 240 injections of which only one is filled with anaesthesia while the rest are filled with medicine that shows no effect on rats.
Now you have 5 rats and 240 injections. Any small dosage of this anaesthesia takes around 24 hours to be effective on a rat. Given the nature of anaesthesia, effectiveness refers to its tendency to make the rat faint. Thus any dosage of this anaesthesia should make them faint in approximately 24 hours. As the rest of 239 injections shows no effect, how will you find that particular injection which has anaesthesia in it using the 5 rats you have, within a period of 48 hours (2 days)?
Now take your time to arrive at a solution and read further once you are done.
Solution
Let’s start solving this by naming the rats as RAT1, RAT2, RAT3, RAT4, and RAT5. Our next step is to label each injection with a number. We are using 5 digit numbers here with 0,1 and 2 being the only digits in them. For example, our labels will look like 11000, 10221, 12011, etc.,
You may have the question- “why 5 digits?…and why only 0,1 and 2?”. This is because we have 3^5= 243 unique ways of arranging the numbers. As we are left with 240 injections, this would give us enough ways of labelling each of them. So far we have labelled them, what are we going to do with these?
Each place in the 5 digit number corresponds to each rat. Let’s take the injection 12110. Here the left-most digit(i.e. 1) corresponds to RAT1 and the right-most digit(i.e. 0) corresponds to RAT5. A rat that is assigned with the number 1 will be given that injection on day 1(i.e at the start of the first 24 hours). Similarly, a rat that is assigned the number 2 will be given that injection on day 2(i.e. at the start of next 24 hours), and a rat which is assigned with the number 0 will not be given that injection at all.
To make this easier, consider an injection labelled 12210. Since the left-most digit belongs to RAT1, a small dosage of this injection will be given to RAT1 on the first day. Second place corresponds to RAT2, so another small dosage of the same injection will be given to it on day 2 because we have the digit 2 in that place. Third place belongs to RAT3 and it’ll be injected on day 2. Fourth place belongs to RAT4 and it’ll be injected on day 1. Fifth place belongs to RAT5 and the corresponding digit being zero, RAT5 will not be given this injection on both days (our RAT5 is so unlucky that it never gets to taste it).
In a similar way, we inject each of those 240 injections by following the label instructions given on them. Here I have made a list of injections received by each rat on both days. Have a quick glance at them if you are still confused with the labelling.
Now that we have injected all these, how do we know which one is the anaesthesia?
Let’s again take the injection labelled 12210 as an example. If 12210 was the only anaesthesia, RAT1 and RAT4 which received its dosage on the start of day 1 should faint after 24 hours. RAT2 and RAT3 which received its dosage at the start of day 2 should faint when the second day ends. Finally, RAT5 which never received this dosage will stay awake. This pattern of fainting will not be repeated in any other injection(if they had been the anaesthesia) as all of them are labelled in a unique way to make the rats faint. Remember, if any of the rats fainted at the end of day 1 itself, it will not be given any further injections as it is already under the influence of anaesthesia.
At the end of 48 hours, by observing the fainting pattern of rats, we can find the anaesthesia by matching the pattern to the label provided on the injection.
Apart from the method I mentioned above, there is another way of arriving at the solution. Split 240 injections among 5 rats and inject each of them with 48 unique dosages. Consecutive injections should be given at a regular interval of 30 mins so that at the end of day 1, all the injections(48×30=240) will be distributed among the rats. Timing should be noted while injecting each time. If any such rat was to faint, we can simply note the time at which it faints and trace back 24 hours to find the injection given at that time, which is our anaesthesia. Unlike the previous method in which we observed a pattern of fainting, here we will observe only one rat getting fainted, as the injection it received wouldn’t have been injected to any other rats.
The issue with using the second method is that the consecutive injection interval of 30 mins is too small to trace back to the right injection as we can’t be sure whether the effect of the drug was delayed. Of course, this method is simpler but works only when we can assure that the drug would show its effect in exactly 24 hours. Hence it is better to go with the first method.
-Vivek Karunakaran
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