NATURAL NUMBERS AND INTEGERS

AQUARIUM.
A cuboidal glass aquarium filled to the brim with water weighs 108 lb. The very same vessel half filled weighs 57 lb. How much does the empty aquarium weigh?

THE YEAR OF SOPHIE’S BIRTH.
In January 1993, Sophie’s age equaled the sum of digits comprised in her birth year. What year was Sophie born in?

A MEASURE OF SUGAR.
With a double pan scale and only four weights of 1-oz, 3-oz, 9-oz, and 27- oz, how does one measure 15 oz of sugar, and then 25 oz?

WHAT DID TOM WRITE?
Tom wrote down two positive integers consisting of the following digits: 1, 2, 3, 4, 5, and 6. Each of the digits appeared in only one of the two numbers, and only once. When Tom added up these numbers, he obtained 750. What positive integers did Tom write?

PUPILS AND GEOMETRY.
The teacher gave her Class-five pupils a difficult geometry problem to solve. It turned out that the number of boys who solved it was greater by one than the number of girls who failed to do so. Which group outnumbered the other: All the pupils that solved the problem or all the girls?

A PECULIAR NUMBER SEQUENCE.
Does a sequence of 11 integers other than zero exists and whose sum of seven successive terms is always positive, whereas the sum of all its terms is a negative number?
Clue: Does an a, b, c three-term sequence exist in which a + b + c < 0, but a+b > 0 and b + c > 0?

ABSENT-MINDED JOAN.
Joan was helping her aunt run a candy shop. When the shop closed after a day’s work, the girl counted all the chocolate bars that remained on the shelves, but due to her absent-mindedness, the number she wrote down in her notebook was missing its final digit. The following morning, her aunt found to her surprise that the number of chocolate bars on the shelves was greater by 89 than the number found in Joan’s notebook. What was the
number Joan should have written down?