I decided I wanted one of those dartboard puzzles for this website, the kind where you are given the scores for each ring of the dartboard, and you have to determine how many darts fell in each ring to get a given total.
I chose 4 different scores for the 4 rings, then I picked a total N that could only be reached one way. It required 10 darts. I noticed that each of the totals N+1, N+2, N+3, ..., N+10 also could be reached only one way, and in each case fewer than 10 darts were needed.
What is the smallest value of N for which this is possible?
Its probably not minimum, but 179 should work. (Assuming whole number scores W<X<Y<Z)
aW+bX+cY+dZ = N
I decided since a+b+c+d=10 a solution could be a=4, b=3, c=2, d=1
Then if W=1, aW=4 so to ensure uniqueness, X=5 and likewise aW+bX=19 so Y=20, adding cY means Z=90. The grand total is 179.
So now N+1 through N+10 can be reached by using a small number of darts
180: d=2
181: a=1, d=2
182: a=2, d=2
183: a=3, d=2
184: a=4, d=2
185: b=1, d=2
186: a=1, b=1, d=2
187: a=2, b=1, d=2
188: a=3, b=1, d=2
189: a=4, b=1, d=2
this can keep going which indicate it's probably not optimal
190: b=2, d=2
My trick to ensure uniqueness pushed N higher and might not have been needed. My values of a,b,c,d, were arbitrary and I might have tried using d=0 instead. I didn't try any of this.

Posted by Jer
on 20240212 15:19:02 