In a certain state, there are citizens of three types: a) a fool considers everyone to be fools, and himself to be smart; b) a modest smart person knows correctly about everyone, but considers himself a fool; c) a confident smart person knows correctly about everyone, and considers himself smart. In the parliament, there are 200 deputies. The prime minister conducted an anonymous survey of the deputies: how many smart people are in this hall right now? Based on the survey data, he was unable to determine the number of smart people. But then the only deputy who had not participated in the survey returned from a trip. He filled out the questionnaire about the entire parliament, including himself, and after reading it, the prime minister understood everything. How many smart people could there have been in the parliament (including the traveler)?
All fools would give the answer "One". If there were three or more wise men in the duma, they would all name numbers not less than 2, and the prime minister would understand everything. Therefore, there could be 0, 1, or 2 wise men. Let's consider all these cases. If there were no wise men, everyone would say: "One". If there was one confident wise man, he would also write "One", and the situation would be indistinguishable from the previous one. If the only wise man was modest, he would answer: "None", and this situation would be distinguishable.

If there were two modest wise men, they would answer: "One", and the situation would be indistinguishable from the first. If there were two confident wise men, they would answer "Two", and the situation would be distinguishable. Finally, if there was one confident and one modest wise man, the confident one would answer "Two", and the situation would also be distinguishable. Thus, there are three indistinguishable scenarios: no wise men, one confident wise man, and two modest wise men. In all these cases, the answer in all questionnaires is "One".

Let's see what answers the tardy duma member would give in each of these situations depending on his wisdom and modesty:

|  | 0 confident wise men  0 modest wise men | 1 confident wise man  0 modest wise men | 0 confident wise men  2 modest wise men |
| :---: | :---: | :---: | :---: |
| Fool | 1 | 1 | 1 |
| Confident | 1 | 2 | 3 |
| Modest | 0 | 1 | 2 |

It is clear that the answers 1 and 2 appear in several cells, so they would not help to distinguish the situations. However, the answers 0 and 3 appear in the table only once and allow for a definitive conclusion. Therefore, the tardy member gave one of these answers. In the first case, there is one wise man in the duma, and in the second case, there are three.

## Answer

One or three.