August 2023 Puzzles
1. I'm thinking of a particular integer.  It is a multiple of 3, 5, and 7.  No digit occurs more than once.  Can you find my number?
With that information there are a lots of possibilities, so let me narrow the problem down with a few more clues.

    a.  The digit in the tens place is a square number.
    b.  The digit in the hundreds place is a cube.
    c.  The digit in the hundred thousands place is both a square an a cube.
    d.  Only the digit in the hundreds place is larger than the digit in the units place.

2. Lucy found that her cow and goat would eat all the grass in a certain field in 45 days, that her cow and goose would eat it in 60 days, but that it would take her goat and goose 90 days to eat it all.  If she turns the cow, the goat, and the goose into the field together, how long will it take them to eat all the grass?  Assume that the grass is not growing each day.
4.  a.  It is a four-digit whole number.
     b.  It is divisible by 5.
     c.  The sum of its digits is 12.
     d.  the sum of its hundreds, tens, and ones digits is 8.
     e.  The sum of its tens and ones digits is 1.
3. In a certain box of candy, the number of caramels is 25% of the number of other candies in the box.  What percentage of the entire box are the caramels?
7.  The number 66 is divided into smaller numbers.  One number is 3 more than twice the other number.  Find the larger of the two numbers.
5. The tens digit of a two-digit number exceeds its units digit by 4.  The number exceeds twice the number obtained by reversing the digits of the original number by 10.  What is the original number?
6.  A man spent three-fourths of his money and then lost three-fourths of the remainder.  He has $6 left.  How much money did he start with?