I have been asked to synthesize the compound (E)-5-methylhex-3-en-2-one:
How can this be achieved using the Wittig reaction?
The Wittig reaction allows for the construction of alkenes from a phosphorus ylide and an aldehyde (or ketone). Shown in the retrosynthetic sense, keeping in mind that there are two possible Wittig disconnections:
Applying this idea to (E)-5-methylhex-3-en-2-one gives the following possibilities:
Which option is chosen often comes down to which gives the most readily accessible starting materials. However, in this case, the stereochemical outcome is critical, and that is controlled by the nature of the ylide used. Looking now in the forward direction, the top option uses an unstabilized ylide, which tend to give the cis-alkene as the major product. The bottom option uses a stabilized ylide, which changes the stereochemical outcome to favor the trans-alkene.
In practice, many chemists would opt for the Horner–Wadsworth–Emmons variant, which gives the trans-alkene as the major product and avoids the production of triphenylphosphine oxide, a byproduct that can be difficult to remove.
The Wittig reaction is used to synthesise double bonds usually from an aldehyde/ketone and an alkyl halide. The following scheme might be one way to achieve your target molecule. Most likely you will have to modify the Wittig reaction, because it is usually (Z)-selective. See also: Why does the unstabilised Wittig reaction selectively form cis alkenes?
If you consider methylglyoxal no good starting material, then you might need to come up with a synthesis for this, too. For the sake of selectivity, you might need protection groups along the way.
This is only one possibility and I am not sure if it actually works, or if the Wittig reaction is the right tool to employ. I am certain, that there are a many other ways of synthesising your target molecule and an extensive literature search would be appropriate.