Consider the following statements: (1) Some blankets are beds. (2) Some pillows are blankets. (3) All beds are pillows. Based on these three statements, which of the given conclusions about blankets, beds and pillows must be logically true?

Introduction / Context:
Questions of this type test basic syllogism and Venn diagram reasoning. We are given relationships between three categories, namely blankets, beds and pillows, and then asked which conclusions must follow. The key idea is that a conclusion must hold in every possible diagram that satisfies the given statements, not just in one convenient example.

Given Data / Assumptions:

• Some blankets are beds.
• Some pillows are blankets.
• All beds are pillows.
• The words some and all are used in the usual logical sense: some means at least one, all means every member of that set.

Concept / Approach:
We translate each statement into set language. The statement all beds are pillows means the set of beds is a subset of the set of pillows. The statement some blankets are beds tells us there is at least one object that is both a blanket and a bed. Whenever some element belongs to two sets, it must also belong to all supersets of those sets. We then check each proposed conclusion against this structure.

Step-by-Step Solution:
Step 1: From all beds are pillows, we know every bed is automatically a pillow.Step 2: From some blankets are beds, there exists at least one item that is both a blanket and a bed.Step 3: Combine Step 1 and Step 2. The item that is both a blanket and a bed must also be a pillow, since every bed is a pillow. Therefore, there exists at least one item that is both a blanket and a pillow. So conclusion 1, some blankets are pillows, is true.Step 4: Since there is at least one bed (from some blankets are beds) and all beds are pillows, those same items are pillows that are beds. Thus conclusion 2, some pillows are beds, is also true.Step 5: The statement some blankets are beds directly implies there exists at least one item that is a bed and a blanket. This is exactly conclusion 3, some beds are blankets, so conclusion 3 is true.Step 6: All three conclusions 1, 2 and 3 hold in every diagram that respects the original statements, so they all must follow.

Verification / Alternative check:
Draw three intersecting circles for blankets, beds and pillows. First, draw the bed circle completely inside the pillow circle to represent all beds are pillows.Next, place at least one common region between blankets and beds, and at least one common region between pillows and blankets. You will always see that some blankets fall inside pillows, some pillows fall inside beds, and some beds fall inside blankets.Try changing the size and overlap of the blanket circle while keeping the constraints valid. In every valid diagram, all three conclusions continue to hold, which confirms our reasoning.

Why Other Options Are Wrong:
Option a is wrong because it ignores conclusion 3, which definitely follows from some blankets are beds.Option b is wrong because it drops conclusion 2, although some pillows are beds is guaranteed by all beds are pillows and the existence of at least one bed.Option c is wrong because it omits conclusion 1, even though we proved that some blankets are pillows must hold.Option e is clearly wrong because not only one but all three conclusions follow.

Common Pitfalls:
A common mistake is to assume some automatically means all, which would incorrectly strengthen or weaken conclusions.Another error is to look at only one or two statements instead of using all given statements together, especially ignoring all beds are pillows when testing conclusions.Some learners also think that some blankets are beds is different from some beds are blankets, but logically they describe the same overlapping relationship.

Final Answer:
The only option that matches the logical analysis is All the three conclusions 1, 2 and 3 follow.