Three statements are given, followed by four conclusions numbered I, II, III and IV. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.

Statements:
Some desks are trays.
Some trays are plates.
Some plates are desks.

Conclusions:
I. All desks are plates.
II. All plates are desks.
III. Some plates are trays.
IV. All trays are desks.

The correct answer is Only conclusion III follows.

To determine which conclusions logically follow from the given statements, we can analyze each statement and its logical conversions step-by-step:

Step 1: Analyze the Statements
1. Some desks are trays: This statement indicates a partial overlap between the set of desks and the set of trays. In syllogisms, the relation "Some A are B" is symmetric, meaning it also implies that Some trays are desks.
2. Some trays are plates: This statement indicates a partial overlap between trays and plates. By symmetry, this also implies that Some plates are trays.
3. Some plates are desks: This indicates a partial overlap between plates and desks. By symmetry, this also implies that Some desks are plates.

Step 2: Evaluate the Conclusions
Let's evaluate each conclusion against the given statements:
- Conclusion I: All desks are plates.
Statement 3 only tells us that "Some plates are desks" (and thus "Some desks are plates"). A "some" statement does not allow us to conclude an "all" statement. We cannot assume that all desks are plates. Therefore, Conclusion I does not follow.

- Conclusion II: All plates are desks.
Similarly, since Statement 3 only guarantees that "Some plates are desks", we cannot conclude that all plates are desks. Therefore, Conclusion II does not follow.

- Conclusion III: Some plates are trays.
Statement 2 states that "Some trays are plates". Since the "some" relation is bidirectional (if some X are Y, then some Y are X), it logically follows that Some plates are trays. Therefore, Conclusion III logically follows.

- Conclusion IV: All trays are desks.
Statement 1 tells us that "Some desks are trays", which implies "Some trays are desks". This does not mean that all trays are desks. Therefore, Conclusion IV does not follow.

Conclusion:
Only conclusion III logically follows from the statements.