There are 6 volumes of books on a rack kept in order (ie vol.1, vol.2 and so…

Concept

In a linear-arrangement puzzle, a relation such as "X is directly to the right of Y" or "Y is on X's left" fixes an exact offset (position of X = position of Y + 1, or -1) between two positions in the row. When several such relations share a common item, chain them together to lock a block of consecutive positions; any leftover relation (adjacency among the remaining items, or a condition tied to one specific original slot, such as a parity check) is then used only to choose among the few ways that block can sit inside the full row.

In such rack- or seat-ordering puzzles, phrases like "X has Y on its right" or "Y is on X's left" are read as immediate placement (X and Y sit right next to each other) - the standard convention for this puzzle style, and the only reading under which the six clues narrow the six volumes to a single order.

Step-by-step application

1. Vol.5 is directly to the right of Vol.2, so Vol.5's position = Vol.2's position + 1.

2. Vol.1 has Vol.3 on its right and Vol.5 on its left, so Vol.3's position = Vol.1's position + 1, and Vol.5's position = Vol.1's position - 1.

3. Chaining these three relations through the shared volumes gives four consecutive slots occupied, in order, by Vol.2, Vol.5, Vol.1, Vol.3.

4. Separately, Vol.6 has Vol.4 directly to its right, so Vol.4's position = Vol.6's position + 1; and neither Vol.4 nor Vol.6 sits in the rack's original third slot (Vol.3's old place).

5. The 4-slot block (Vol.2, Vol.5, Vol.1, Vol.3) and the 2-slot block (Vol.6, Vol.4) together must fill all six slots without overlapping, so the 4-slot block can only start at slot 1, 2, or 3.

6. Starting the block at slot 2 leaves slots 1 and 6 for Vol.6/Vol.4 - these are not adjacent to each other, so this start fails.

7. Starting the block at slot 3 leaves slots 1 and 2 for Vol.6, Vol.4 (adjacent, and not slot 3, so this pair is fine), but then slot 5 is occupied by Vol.1, an odd-numbered volume - which breaks the condition that an even-numbered volume must sit in the rack's original fifth slot.

8. Starting the block at slot 1 leaves slots 5 and 6 for Vol.6, Vol.4 - adjacent, and neither lands on slot 3 - and slot 5 then holds Vol.6, an even-numbered volume, satisfying every condition.

9. This gives the final order: Vol.2, Vol.5, Vol.1, Vol.3, Vol.6, Vol.4.

Cross-check

Verifying this order against every original clue: Vol.5 sits immediately right of Vol.2 - holds; Vol.6 sits immediately left of Vol.4 and neither is in the rack's original third slot - holds; Vol.1 has Vol.3 on its right and Vol.5 on its left - holds; and the volume in the rack's original fifth slot is Vol.6, which is even - holds. All conditions are satisfied only by this order.