c4countdown

My dad and I were spotting on Newport station last week and it got to about 1am when we needed to keep ourselves occupied to avoid being cold and sleepy. My dad put forward a challenge. The challenge was using the numbers 1, 2, 3, 4, 5 and 6, what is the smallest target in Countdown that cannot be obtained? (I have a sneaky suspicion this has been discussed before but I couldn't remember it). It got really interesting when we used split multiplication, e.g. for 6x5x4 (120), you have a 3, 2 and 1 left over so can easily get 121, 122, 123, 124, 125, 126, 127 and 119, 118, 117, 116, 115, 114, 113 but also if you add the 1 before multiplying some numbers, e.g. (6+1)x5x4, or (6x5+1)x4. This generated so many more targets very quickly.

Anyway, without cheating (google etc) and without using computer aids, can anyone come up with the smallest number that isn't possible. It may take a couple of hours. We thought we found a solution but I used an online solver and it was in fact possible, albeit with only one solution.

Good luck!

This got me thinking about the generalised problem of finding the smallest target that can't be made from the numbers 1,...,n. I worked it out in my head up to n=4 and then had a look around, and sure enough someone has computed it up to n=13: http://www.research.att.com/~njas/sequences/A060315 (nb spoilers for original puzzle).

I was thinking more generally - which are the "best" targets and the "best" selections? So the best selections would be ones where you can make the most targets and the best targets would be the ones that you can make from the most selections.