Sticky fingers
Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?
Age
14 to 16
Challenge level
Problem
Ruth has glued 2009 unit cubes together to form a cuboid.
She opens a pack containing 2009 stickers, and she has more than enough to place one sticker on each exposed face of each unit cube.
How many stickers does she have left?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $763$
Three positive integers that multiply to make $2009$ are:
$1 \times1 \times 2009$
$1\times 7\times 287$
$1\times 41\times 49$
$7\times 7 \times 41$
$1\times1\times2009$ has faces $1\times1$ (two faces), $1\times2009$ (four faces). Not enough stickers for the $2009$ face.
$1\times 7\times 287$ has a face which is $7\times287=2009$ - not enough stickers.
$1\times 41\times 49$ has a face which is $41\times49=2009$ - not enough stickers.
$7\times7\times41$ has faces $7\times7$ (twice) and $7\times41$ (four times)
Surface area $2\times7\times7+4\times41\times 7 = 1246$
This leaves $2009-1246=763$ stickers left over.