I think this attempt is closest to the "five years old" level:
So you have 23 questions, and on each you can either answer yes or no. You goal is that your friend "transmits" you some information so that you get the most of the answers right. As the problem is set up, what he can transmit to you is just an exact second of a 90 minutes span, that is one number between 0 and 5399.
Having 23 questions, every with 0 or 1, gives 23 bits of information to pass. 4096 is 2 to 12 th power, so the friend can pass you only 12 complete bits.
Now the major point: you're lucky that you don't care which answers exactly you get right. You need 23 bits passed to get the exact 100% right solution. But you don't care. It turns out, if you cleverly write on the paper 4096 different "solutions" before the test and your friend transmits you which one of these 4096 you should take to fill out the form, the mentioned cleverness can guarantee you that you get N answers right, you just don't know which specifically in advance and you don't care as you prepare these 4096 solutions. The friend must also know the same table, and has to select one of the prepared 4096 solutions only once he discovers what the "100%" solution is.
The remaining question was how many answers you can be sure you can get right (N), and it turns out, N is 20.
Why? If you carefully construct each of these 4096 solutions to be "as different" from all the other 4095 solutions as possible, it can be proven that then 20 is the result.
So you have to take to the test a list with 4096 longer binary numbers, carefully prepared, your fried can transmit you which one to write through the time of his leaving of the test, and you'll have 20 answers right. Or you can train yourself to perform a special algorithm to produce a pattern of 23 answers from the time and the friend to do the opposite.
The math details are, thanks to allenz, here (not on "five years old" level): https://en.wikipedia.org/wiki/Binary_Golay_code There are also some algorithms. The 20 in the solution can be figured out from the sentence "G23 is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space." Which for our task means that not more than 3 answers will be "wrong" from 23 answers known to your friend, if the whole trick is performed correctly, therefore 23-3=20.
And of course, this kind of cleverness (but used for error correction, that is, to allow sending more data than minimally necessary, but where some of the data is lost before it reaches the other side and everything still works) is actually built-in in the modern communication and data storage equipment, but we, typically, just reap the benefits and don't think of it.
On another side, human languages also evolved to contain redundancies, so the evolution is very capable to find the acceptable solutions for the message encoding too.
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1) This way you haven't used all 5400 possible values, but now your can split the remaining thousand-something seconds between you and your friend, for him to prepare the transmission, and for you to fill the form after you received the code but before the test is over.