The presence of the alpha-beta unsaturated ketone functionality (see note to the right) suggests the use of an aldol condensation to form the product. This would simplify our target to A. In the forward sense, formation of the enolate of the methyl ketone at the top, subsequent attack of the ketone in the lower left, and loss of water would result in the desired molecule.
Remember: when performing retrosynthetic analysis, it is important to consider the merit of the reactions in the forward sense. In this case, while one may worry about possible side reactions (by formation of another enolate), these aldols are in equilibrium, and the thermodynamic stability of the six-membered ring would favor the desired product. This step is indeed feasible in the forward sense.
Examining A, one observes a 1,5 diketone relationship embedded in the structure. This suggests that a Michael reaction could be used to form A from B and C.
Note: This retrosynthetic step would be a more sensible choice than the one shown below.
Why is this? Think about it and then check the answer...
B is a beta keto ester, which calls to mind the Claisen ester condensation...
...and C is an alpha beta unsaturated ketone which can be easily formed in an aldol reaction.

Again, remember to consider what is happening in the forward sense. In this case, this mixed aldol reaction is acceptable since even though there are two different aldehydes involved, only one has acidic protons, and only one enolate can be formed.

I leave the forward synthesis to you. If you have any problems determining what reaction conditions to use, consult your notes and Professor Wintner's problems.

Question: The combination of B and C to form the desired target of question four in two steps is an example of a special, named sequence of reactions. What is the name for this Michael addition and subsequent intramolecular aldol condensation? The answer can be found in the solutions to question 6 (near the bottom of the page).

One last thought on problem 4. If the target was just like the one given, except without the ester group next to the ketone at the top of the molecule, the synthetic pathway would be the same, with the added last step of hydrolysis of the ester and subsequent decarboxylation. Retrosynthetically, the first step would be the installation of this ester (since this allows us to perform the Michael reaction using the stable enolate derived from B). Be able to recognize when it is appropriate to retrosynthetically add this ester group! (Another example is alkylations of enolates.)
More answers...
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