Miscellaneous Puzzles

These are the puzzles which don't belong in any of the other sections. Some are easy, some are harder.

1. A Question of Time

The hands of a clock obviously lie one above the other at 12 noon. At preciesly which other times will the position of the two hands coincide once more?

2. Guess my Age

"Two days ago little Lucy was only 7. However, next year she will be 10."

When is her birthday, and what is the date today?

3. The Impossible Sum

Make 24 using two '8's and two '3's. You must use all four digits, and they must be used as single digit numbers. They can be combined only using the four natural arithmetic operators (+, –, ×, ÷) together with parentheses (brackets). There is no restriction on how many of each operator is used.

4. Digital Dexterity

Using the ten digits 0-9 once each, make a ten-digit number N = [d1d2d3...d9d10] (in which each di represents a single digit) with the following property:
For each n in {1,2,3,...10} the number [d1d2...dn] formed from the first n digits of N should be exactly divisible by n.

5. Time and Again

You are given three sand-filled hour glasses (egg timers) that each measure exactly 12 minutes of time. Find a way, using just the three glasses, to measure a period of 6 minutes of time. What other periods of time can you measure? Would it help if you had more than three hour glasses available?

Hint: You may assume that in each glass the sand falls through the narrowing at a constant rate, regardless of the amount of sand remaining above. This is a reasonable approximation for the flow of a granular medium.

6. Sweet Transfers

You have three jars each contining a number of sweets. On each turn you must take one sweet from each of two of the jars, and place both sweets into the third jar. Prove that if the total number of sweets is not a multiple of three, then it is always possible to end up with all the sweets in one jar, however they are distributed to start with. What happens if the initial number is a multiple of three?