Question

Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower $94$ bricks tall. Each brick can be oriented so it contributes $4'',$ $10'',$ or $19''$ to the total height of the tower. How many different tower heights can be achieved using all $94$ of the bricks?

Answer

Answer: 465

Concepts:Diophantine Equation parity counting integers in a range

Difficulty rating: 2270

Small Hint:

Start with all bricks contributing $4$ inches and count possible increments

Big Hint:

If $c$ bricks contribute $19$ inches, the remaining increments form a parity interval as the number of $10$ -inch bricks varies

Solution:

If $b$ bricks use height $10$ and $c$ use height $19,$ the total is $376+3(2b+5c),$ where $b,c\geq0$ and $b+c\leq94.$ For fixed $c,$ the value $v=2b+5c$ runs by twos from $5c$ to $188+3c.$ The even- $c$ intervals cover every even $v$ from $0$ through $464,$ as well as $470,$ missing only $466$ and $468.$ The odd- $c$ intervals cover every odd $v$ from $5$ through $461,$ as well as $465$ and $467,$ missing only $1,$ $3,$ $463,$ and $469.$ Thus there are $471-6=465$ distinct heights.

1994 AIME Problem 11

Answer: 465

Difficulty rating: 2270

Small Hint:

Big Hint:

Solution:

Problem 11 in Other Years