The question is to propose a synthesis of indane from benzene. I started by the following. The black box is what I am not sure about. I know that the stable carbocation should be the middle one, this is why I am not sure. I need help here.
To prepare indane (6) from benzene, make the procedure as simple as possible and amenable to scale-up. Propiophenone (2) is readily available by Friedel-Crafts acylation of benzene with propionyl chloride (1). Removal of the carbonyl group of propiophenone and the creation of a carboxyl moiety can be achieved via the Willgerodt reaction to produce amide 3. Vigorous acid or base cleavage of the amide is possible but nitrous acid affords a mild route to carboxylic acid 4. One can use the AlCl3 F-C route again but hot polyphosphoric acid is a clean method for the intramolecular acylation without having to form the acid chloride. Finally, Clemmensen reduction of indanone (5) gives indane (6). Wolff-Kishner reduction is also a good option for the final step.
Below is a suggested route for synthesizing indane starting from benzene and allyl bromide. I appreciate that this may not be a desirable answer for a school question but it is how one could perform the synthesis in real life and is based on high yield reactions that have been previously reported (see references, also note that this may not be the most efficient way):
References: 1) Synlett, 1999, No. 8, 1245. 2) J Org. Chem. Vol.78, No 19, 9772. 3) J Org. Chem. Vol.57, No 23, 1992. 4) Molecules 2014, Vol 19, No 5, 5599. 5) Catalysis letters 2011, Vol 141, No 4. 507.