WASHINGTON (AP) _ A nimble-minded Illinois boy with a gift for crunching numbers in his head won a medal and an \$8,000 college scholarship Friday in a national mathematics bee.

William Engel, 12, of Aplington Middle School in Polo, Ill., also won a personal computer and a week's stay at the U.S. Space Camp in Huntsville, Ala., as first-place winner of the 11th annual Mathcounts National Competition.

He was one of 224 seventh- and eight-graders from across the country and from American schools overseas participating in the event.

William was declared the champion after calculating the volume of a scale model of the Washington monument using only scratch paper. He paused only a few seconds before giving the right answer - 8,413 cubic inches.

William said he spent a half-hour to an hour a day studying to prepare for the competition.

William is a good student, but he's also blessed with a natural talent for mathematics, his father said. ''We noticed he was special pretty early,'' said his father, Eldon Engel, a mailman. ''It's God's gift.''

''He has an unbelievable ability to do problem-solving,'' said Deborah Cheek, who has been the boy's math coach since the fourth grade. ''He can do it fast. He has an innate gift for calculation.''

Second-place winner Michael Schulman of Valley Village, Calif., said math isn't even his favorite subject.

''I didn't really have a large interest in it before now,'' said Michael, 13, who attends Walter Reed Middle School. He won a \$6,000 scholarship.

Kevin Lacker, 13, of Sycamore Junior High School in Cincinnati won the third prize, a \$4,000 scholarship.

Pennsylvania, California and Texas ranked first, second and third, respectively, in the team competition. Each team had four members.

Some of the questions asked in the final round were:

-A model of the Washington Monument is built to a scale of 1 inch to 5 feet. Given that the volume of the Washington Monument is 1,051,625 cubic feet, how many cubic inches are in the volume of the model? Answer: 8,413 cubic inches.

-The first row in a theater has 26 seats. Each of the rows behind has two more seats than the row immediately in front of it. How many seats are in the first 10 rows? Answer: 350.

-Find 103 times 97 plus 9. Answer: 10,000.

-The perimeter of a regular hexagon is 194.4 centimeters. How many centimeters are in the perimeter of a regular octagon whose side has the same measure as the side of the hexagon? Answer: 259.2 centimeters.

To qualify for participating in Friday's national competition, student ''mathletes'' advanced through local and state competitions involving more than 300,000 students. Thirty-seven girls and 187 boys took part in the finals.

They represented all 50 states, the District of Columbia, the five U.S. territories and schools that the Defense Department and the State Department operate overseas.

The competition is sponsored by Mathcounts, an Alexandria, Va.-based coaching and competition group promoting math achievement. The group is supported by the CNA Insurance Cos., the National Society of Professional Engineers, Cray Research Foundation, General Motors Foundation, Intel Foundation, Texas Instruments Inc., the National Council of Teachers of Mathematics and the National Aeronautics and Space Administration.