Four Fours

I stumbled upon this little brain teaser a few months ago. Since then, I've kept a sheet of paper near my computer that I've added to when something strikes me, and I think I've finally completed it. Here's the puzzle:

1. Start with four fours: 4 4 4 4


2. Add mathematical operators between and around these four fours so that you end up with equations that equal the numbers 1 through 20.

Hint: You'll have to use six different operators, as well as parentheses, to accomplish this.

Here are my answers. Some of these (the first one, is a good example) have more than one possible answer. I went with the one that involves the fewest HTML codes. (And, yes, some of the parentheses are redundant because of the order of operations, but I think they're also easier to read.)

See if you can figure it out, and then click the link below to see my answers.

1. (4 + 4) ÷ (4 + 4)


2. 4 × (4 ÷ (4 + 4))


3. (4 + 4 + 4) ÷ 4


4. (√4 + √4) × (4 ÷ 4)


5. ((4 × 4) + 4) ÷ 4


6. √4 + 4 × (4 ÷ 4)


7. (4 + 4) – (4 ÷ 4)


8. 4 + 4 + 4 – 4


9. 4 + 4 + (4 ÷ 4)


10. 4 + 4 + 4 – √4


11. (4! ÷ √4) – (4 ÷ 4)


12. (√4 + √4 + √4) × √4


13. (4! ÷ √4) + (4 ÷ 4)


14. 4 + 4 + 4 + √4


15. (4 × 4) – (4 ÷ 4)


16. 4 + 4 + 4 + 4


17. (4 × 4) + (4 ÷ 4)


18. 4! – (√4 + √4 + √4)


19. 4! – 4 – (4 ÷ 4)


20. (4 × 4) + (√4 × √4)

Meeting notes

I was pretty tired today because back pain kept me tossing and turning last night. In an afternoon meeting, I even started counting sheep. Here are my notes from the meeting:
Okay, so one of them is really a wolf in sheep's clothing. Either way, it didn't help me fall asleep this afternoon.

FAIL

One of my new favorite blogs: FAIL Blog.

Some happy shots from that blog:

IWS Season Ender

I think that our final concert went well. I got very lucky with the PowerPoint presentation — especially considering that the conductor unknowingly started the second movement before I had even pulled it up on the screen. I've got nothing but compliments about it.

The silent auction, though, was pitiful. We didn't even have two dozen items up for auction. (Last year, we had over 30. But we also had a larger audience.) Last year we made about $1,000; this year we didn't even break $400.

I just don't know what to do to get people in the band to put in some extra work. I know I need to help create a sense of ownership among the band-members, get them invested, so that they will stop just expecting things to happen the way they're supposed to. Most of the people just come and play, and the behind-the-scenes machinations that keep the ensemble going are just SEP: somebody else's problem.

I think we need some measurable goals to work toward. We need both monetary and musical goals, and some way to report how close we are to achieving those goals. That way, it won't just be a string of individual concerts, but a single continuing season. But how do we do that?