Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."
Part A. Write down the positive integers, cross out every second, and form the partial sums of the remaining.
1 2 3 4 5 6 7 8 9 10 11
1 4 9 16 25 36
Does the pattern of squares continue?
Part B. As before, but cross out every third, form partial sums, then cross out every second and for a second partial sums.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 7 12 19 27 37 48 61 75 91
1 8 27 64 125 216
Does the pattern of cubes continue?
(In reply to
Notes by Jer)
I would expect the algorithm works for all powers, say for tenth powers start with erasing every 10th number, take the partial sums, then erase every 9th number, etc.
To test this I wrote a UBASIC program to try it out. The output writes the successive sequences with asterisks preceding the values to delete for the next row.
10 input "B^E B=";B
20 input "B^E E=";E
30 Size=E*(B-1)+2
40 dim V(Size)
50 for X=0 to (Size-1):V(X)=X:next X
60 Size=Size-1
70 for P=E to 2 step -1
80 J=1
90 for I=1 to Size
100 if (I@P)=0 then
110 :print " *";V(I);
120 :else print V(I);:V(J)=V(J-1)+V(I):J=J+1
130 next I
140 Size=Size-(B-1):print
150 next P
160 for X=1 to B:print V(X);:next X
The results for the first five 10th powers:
1 3 6 10 15 21 28 36 * 45 56 68 81 95 110 126 143 161 * 180 201 223 246 270 295 321 348 376 * 405 436 468 501 535 570 606 643 681 * 720 761
1 4 10 20 35 56 84 * 120 176 244 325 420 530 656 799 * 960 1161 1384 1630 1900 2195 2516 2864 * 3240 3676 4144 4645 5180 5750 6356 6999 * 7680 8441
1 5 15 35 70 126 * 210 386 630 955 1375 1905 2561 * 3360 4521 5905 7535 9435 11630 14146 * 17010 20686 24830 29475 34655 40405 46761 * 53760 62201
1 6 21 56 126 * 252 638 1268 2223 3598 5503 * 8064 12585 18490 26025 35460 47090 * 61236 81922 106752 136227 170882 211287 * 258048 320249
1 7 28 84 * 210 848 2116 4339 7937 * 13440 26025 44515 70540 106000 * 153090 235012 341764 477991 648873 * 860160 1180409
1 8 36 * 120 968 3084 7423 * 15360 41385 85900 156440 * 262440 497452 839216 1317207 * 1966080 3146489
1 9 * 45 1013 4097 * 11520 52905 138805 * 295245 792697 1631913 * 2949120 6095609
1 * 10 1023 * 5120 58025 * 196830 989527 * 2621440 8717049
1 1024 59049 1048576 9765625
Edited on October 5, 2017, 2:16 pm
re: Notes
