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Depending on the person, you can survive more or less for about a month on water alone. Your mental and physical performance will be suboptimal, to say the least. If you have a fast metabolism and no fat stores, you will starve to death faster than a fat person. But even the fattest person won't last much longer than a month with no food. You can starve to death and still be fat, because your body can't synthesize ATP from its own proteins/fats fast enough and without causing serious damage to itself. You would die from organ failure and/or suffocation before your fat got used up.

As for living on dates and milk the whole rest of the year, I don't know if dates and milk contain all the nutrients you need to be healthy. Between the two of them, they probably do. Even if not, I suspect you could survive for a very long time eating only dates and milk. It was possible for people to survive on bread and water for long periods of time in the days before refined flour, because the bread still contained most of the nutrients from the plant. I think dates and milk would provide better nutrition than just bread.