You can throw as many darts at the board below.   Some total scores are impossible to obtain -- for example, all numbers less than 5, as well as 6, 8, and 9.

What is the highest whole number score that is impossible to obtain?

Extra Credit: Can you come up with an algorithm or rule to determine the largest impossible score for any two whole numbers on the dartboard?



 

Solution to the Problem:

The answer is 23.

Begin by writing down numbers which are possible:
5, 7, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, ...
Once you get five scores in a row (like we did from 24 to 28), you can just add 5 to those scores to get the next five scores.   Then adding five to those scores gives you the next five scores.   From here on out, all scores are possible.   Hence 23 was the largest impossible score.

If the dartboard had different numbers on it, the following rules apply:
  1. If both numbers are even, all odd numbers are impossible.
  2. If one number is a multiple of the other, all numbers that are not multiples of the smaller number are impossible.
  3. Otherwise, if one number is a and the other number is b, then the largest impossible number
      is     ab - a - b.
I posted a similar problem back in 1998 involving Chicken McNuggets.   At McDonald's, you can order McNuggets in boxes of 6, 9, or 20.   What is the greatest number of McNuggets that you can not order?
Click here for the answer to the McNuggets problem



Correctly solved by:

1. Sagar Patel *** Brookstone School
Columbus, Georgia
2. Shaan Arora *** Brookstone School
Columbus, Georgia
3. Jim Arrison *** Norristown, Pennsylvania
4. Keith Mealy *** Cincinnati, Ohio
5. Kunal Singh Kendriya Vidyalaya School
Happy Valley, Shillong, India
6. Larry Schwartz *** Norwalk, Connecticut
7. Tristan Collins Virginia Tech
Blacksburg, Virginia
8. David & Judy Dixon Bennettsville, South Carolina
9. James Alarie *** University of Michigan -- Flint
Flint, Michigan
10. Richard Johnson *** La Jolla, California
11. Andrew Dau South O'Brien High School
Gaza, Iowa
12. Dana Moats Ft. Eustis, Virginia
13. Kerry Barwick ----------
14. Magdy Essafty Alexandria, Egypt
15. Lauren Champion Brookstone School
Columbus, Georgia
16. Slayton Gilmore Brookstone School
Columbus, Georgia
17. Javier Johnson Washington Township High School
Sewell, New Jersey
18. Nicholas Caligaris *** Brookstone School
Columbus, Georgia
19. Troy Amos Brookstone School
Columbus, Georgia
20. Teresa Franklin Dandridge, Tennessee
21. Mark Turley John Handley High School
Winchester, Virginia
22. Leah Vahjen Brookstone School
Columbus, Georgia
23. Jessica Thorpe Brookstone School
Columbus, Georgia
24. Ben Bassett Washington Township High School
Sewell, New Jersey
25. Jessie Biggs John Handley High School
Winchester, Virginia
                        *** Also answered the extra credit