What decimal digit occurs most often in the sequence of factorials from 77777! to 77877! inclusive?

This can be solved by logic. You do not need to compute any factorials.

Alex has before him some positive whole numbers, each consisting of a single digit, which may be repeated. The digit is different for each number, and the number of times it is repeated is also different for each number.

The sum of Alex's numbers is a number in which each digit is larger than the digit on its left, and it is the largest number for which this is possible, given the constraints described above.

What is the sum of Alex's numbers?

__Note__: Adapted from Enigma number:1765 which appeared on "New Scientist" in 2013.

D be a point inside triangle ABC.

The following are given:

Angle DAC = 30° and Angle DCA = 40° , AB = BD + DC, BD = AC

Find angle DBC.

Five married couples are out for a pleasure cruise on a yacht when the boat hits a rock and sinks. They are left stranded on a small island 6 miles from shore. Luckily, there is a rowboat moored on the island large enough to hold 3 people, and the rowboat is equipped with 3 sets of oars and oarlocks.

Each of the men can row the boat at 3 mph, and each of the women can row it at 1 mph. If two or more people row together, their speeds are added. That is, 2 men would row at 6 mph, a couple would row at 4 mph, etc.

The husbands are all very jealous, and will not allow their wives to be with any of the other men unless they are also present, either in the boat, on the island or on shore. Several women may be together, but if any man is with them, all of their husbands must be there. (When the boat is loading or unloading, all of the people, both in and out of the boat, are considered to be together.)

In the shipwreck, 7 of the people suffered neck injuries, and cannot turn their heads to see where the boat is heading, so one of the 3 uninjured people must be in the boat whenever it is being rowed.

What is the least time required to get everybody safely to shore? (You can decide which people are injured.)