When successive powers of 5 are divided by 7, the remainders form a repeating series:

5¹ / 7 = 0 remainder 5

5² / 7 = 3 remainder 4

5³ / 7 = 17 remainder 6

5⁴ / 7 = 89 remainder 2

5⁵ / 7 = 446 remainder 3

5⁶ / 7 = 2232 remainder 1

5⁷ / 7 = 11160 remainder 5

5⁸ / 7 = 55803 remainder 4

5⁹ / 7 = 279017 remainder 6

5¹⁰ / 7 = 1395089 remainder 2

…

The 999,999th term of the series is 6.

It’s 1 in 5. Six possible pairings can be drawn from the hat:

red 1 / red 2
red 1 / white
red 2 / white
red 1 / blue
red 2 / blue
white / blue

We know that the last has not been drawn. That leaves 5 possible combinations, so the possibility of red 1 / red 2 is 1 in 5.

“Many people cannot accept that the solution is not 1 in 3,” writes Erwin Brecher, “and of course it would be, if the balls had been drawn out separately and the color of the first ball announced as red before the second had been drawn out. However, as both balls had been drawn together, and then the color of one of the balls announced, then the above solution, 1 in 5, must be the correct one.”

1. Qh7 (K moves) 2. e8=Q#.

Turn over both hourglasses. When the 4-minute glass has run out, turn it over. When the 7-minute glass finishes, turn it over and let it run until the 4-minute glass has run out. At this point we’re at the 8-minute mark and the lower bulb of the 7-minute glass contains 1 minute’s worth of sand. Turn it over to measure the final minute.

It’s better not to spin it. Two of the six chambers are loaded, so my chance of surviving was 4/6. But you know that the hammer fell on an empty chamber. Of the four empty chambers, only one is followed by a loaded one. So if you simply pull the trigger, you have a 3/4 chance of surviving. That’s better than 4/6, so you’re better off not spinning the cylinder.

The words PULL and PUSH were painted on opposite sites of the door.

1. b8=N! Kxe8 2. Qg8#

If he runs toward the train, he’ll cover the remaining 3/7 of the trestle and meet the train at the far end. This means that if he runs away from the train, he’ll have covered 3/7 of the trestle (with 1/7 to go) when the train reaches the far end. He’ll then run the remaining 1/7 while the locomotive crosses all 7/7 and passes him. So the train is traveling at 7 times his speed, or 140 kph.

The waitress has asked, “Do you all want coffee?” In saying “I don’t know,” each of the first five logicians is saying, “I want coffee, and at this moment it does appear possible that all six of us do, but we don’t yet have enough information to say so definitively.”

The sixth logician doesn’t want coffee, so he’s able to say for certain that not everyone in the group does. So the waitress serves coffee to the first five logicians.

Yes. He puts on both pairs of gloves and performs the first operation. He puts aside the outer pair of gloves and performs the second operation. Then he turns the first pair inside out and puts them on over the now-used second pair. This puts the two non-sterile sides in contact while keeping his hands in their sterile state and making the remaining sterile side (formerly the inside) of the first pair of gloves available for use in the third operation.