Sometimes the lessons we teach keep on giving long after we think our classes are over. As you may know from my previous posts (here and here), I used an adaptation of the Four 4s activity to try to foster some creativity in my classes this Spring.

Basically, you are allowed four 4s, no more and no less, and no other digits, but any mathematical operations you want. From this, create every integer from 1-100. Of course, you can go far beyond that, but this seemed like a good place to start. Two of my students employed an unexpected approach, seeking ways to find all numbers 0-25 using a single four, thereby proving to themselves that the entire 1-100 table was solvable–often in multiple ways–without actually needing to write all the solutions. It was a nice existence proof.

OK, most of their solutions were VERY complicated and the few they submitted ended up being sniped by others, but that wasn’t their goal. They wanted to know a solution was possible. Two weeks ago, one of them, P, sent me an email describing how he leveraged his initial work to find solutions to the *next 11 missing integers* in Integermania’s Four 4s list. Talking to P the next day, I found out that he accomplished all 11 in about half an hour. To put this in perspective, I think Integermania has been running this list since early 2006. Admittedly, most participants probably lost interest and stopped submitting entries long ago, but I still found P’s ability to find so many solutions so quickly to be pretty powerful. The list has been designated “mostly inactive”; we’ll submit them anyway…

Here’s P’s unedited email (other than some LaTeX conversions by me). If you need it, the functions he uses are detailed in the middle of my 2nd Four 4s post linked above.

Hi Dr. Harrow, I was in your room at about 4:00 and live about 30 minutes from school, got home, had a snack, opened integer mania. Once I scrolled through all the pages and found that the next missing number is 1138:

Realize .

Using my list of 25, we can get 1140 using only three fours, then subtract 2 to get

Now, , so

Which completes their list through 1200 (actually 1206). Just to emphasize how easy this is, find a way to get from 1200 to 1207 using only two fours. Oh man…

Thanks, P

PS: The highest exquisiteness level so far is the dude who made a googolplex (last page) with a 5.8. I might have gone a little past that…

Pretty impressive, I thought. As for P’s exquisiteness, here’s what I computed

- 1138 = Base 7 + 5 surcharges = Level 8.0
- 1139 = Base 7 + 6 surcharges = Level 8.2
- 1142 = Base 7 + 5 surcharges = Level 8.0
- 1143 = Base 7 + 5 surcharges = Level 8.0
- 1159 = Base 7 + 6 surcharges = Level 8.2
- 1162 = Base 6 + 6 surcharges = Level 7.2
- 1163 = Base 6 + 6 surcharges = Level 7.2
- 1169 = Base 7 + 7 surcharges = Level 8.4
- 1171 = Base 6 + 6 surcharges = Level 7.2
- 1183 = Base 6 + 5 surcharges = Level 7.0
- 1193 = Base 6 + 4 surcharges = Level 6.8

That these are pretty high levels relative to the rest of the list is totally irrelevant, in my opinion. P has found a simple way to prove *existence *of a solution. Often, a solution’s existence is enough to spur on investigation of more elegant answers. P broke through. Knowing that answers are *possible*, he challenges others to follow up with smoother results.