It is not enough to have a good mind. The main thing is to use it well. —René Descartes
This logic puzzle reportedly appeared in Omni Magazine decades ago, and it's an astounding one because it seems to lack sufficient information for a solution. This deceptively simple puzzle captures the essence of problem-solving under multiple constraints and the exploration of non-obvious implications, the kind of cognitive processes discussed in the previous article. In addition, it unfolds in a charming diplomatic environment where imagination makes way for a cold, symmetrical logic, riddled with abstract beauty!
A Diplomatic Puzzle
During an International Summit, each of the Commissioners and the accompanying Delegates from the 27 country members, took part in the inaugural session. There were 54 people in all; two from each country (one Commissioner and one Delegate). Before the start of the session, some of the participants shook hands with each other, although no Commissioner shook hands with his/her Delegate, and no two people shook hands more than once. After the session, the Commissioner from Germany asked everybody how many times they had shaken hands. All the participants (excluding the German Commissioner) answered and the numbers they gave were all different. How many times did the German Commissioner shake hands?\(^{1}\)
Solution
First of all, we can determine the number of all possible
handshake counts each person can have. Each participant cannot shake hands with themselves, and each one of them did not shake hands with his/her country's partner. Therefore, the maximum number of handshakes a participant may make is \(54-2=52\). As a result, the range of all possible numbers of handshakes for any given person goes from \(0\) to \(52\). Note that each person, besides the German Commissioner, reported a different number (i.e., a distinct integer), and since handshake counts must be integers between \(0\) and \(52\), the only possible way for this to occur is if the \(53\) people's reported counts are in
one-to-one correspondence with the set \(\{ 0,1,2,...,52 \}\), which has extacly \(53\) distinct elements.
Now, consider what occurs whenever a given person shakes hands a specific number of times. For instance, suppose a person shook hands \(0\) times, then their corresponding partner from the same country must have shaken hands \(52\) times since the person with \(0\) handshakes shook hands with nobody, and their partner can neither shake hands with themselves nor can they shake hands with their partner who shook none.
Why, you may ask, does this follow? Well, we know that someone reported a count of \(0\), and we also know that someone must report a count of \(52\). Who could possibly report \(52\)? Only the person who shook hands with every participant they were allowed to — namely, the partner of the person who shook hands with nobody.
If this person had shaken hands with fewer than \(52\) people, then no one else could report \(52\). Any other person attempting it could shake hands with, at most, \(51\) people (they cannot shake hands with themselves, the person who shook hands with nobody, or their own partner). But \(52\) must appear somewhere among the answers, which is a contradiction, and thus impossible!
The same reasoning shows that the person who shook \(1\) hand must be paired with the person who shook \(51\), the person who shook \(2\) hands must be paired with the person who shook \(50\), and so on and so forth. We may express this situation as a
set of ordered pairs,
$$\{ (0,52), (1,51), (2,50), ..., (26,26)\}$$
We are now on the threshold of a solution because the same piece of information that led us to deduce the number of possible values of handshake counts is the one on which our spectacular finale depends (though, to be sure, a very diplomatic one). Observe that among the \(27\) possible ordered pairs there are \(26\) pairs \((a,b)\) where \(a\neq b\), which indeed correspond to the given responses. Consequently, the only pair left, in which both numbers are the same, must correspond to the German Delegate and must be paired with the German Commissioner, that is, \((26,26)\). Hence, the German Commissioner shook hands \(26\) times!
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