The Dart Booth

Throw three darts. Cross off the score.

Every dart sticks in the board. Try to cross off all twelve numbers on the strip below.

4 2 1
— — —
Tap a ring to throw your first dart.

The number strip

0 of 12 crossed off

Three numbers will not budge

You found all nine that are possible. 1 and 2 are easy to explain: three darts always land, and the smallest ring is worth 1, so nobody can score less than 3.

Now try this way of looking at it. Every dart starts out worth 1, so you begin with 3 points for free. Moving a dart to the middle ring earns 1 more. Moving it to the bullseye earns 3 more. Three darts, so three upgrades.

To reach 11 you would need 8 extra points. But upgrades only come in 1s and 3s: the best you can do is 3+3+1 = 7, and the next step up is 3+3+3 = 9. Nothing lands on 8. You would need 3+3+1+1, and that is four upgrades — one dart too many.