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 Prime Hopscotch (Posted on 2010-01-06)
Replace the numerals 1 through 8 with ever increasing prime numbers, always using the next lowest possible that is available1 to fulfill the criteria on the left. Then do the same for the right.

Present a series for the left, and one for the right.

 7 + 8 = Square 6 = Prime 4 + 5 = Square 1 + 2 + 3 = Square 7 + 8 = Cube 6 = Prime 4 + 5 = Cube 1 + 2 + 3 = Cube

If it was required that the Right set required "1" to be the next Prime following on after that used for the "8" in the Left set, what might the Right set read, if indeed it is possible?

1. Note, "always using the next lowest possible that is available" means that if it is next on the list it cannot be dismissed unless it is the last of a group of two or three and will not fulfill the criterion. Only then may you advance to the next.

 See The Solution Submitted by brianjn Rating: 4.0000 (1 votes)

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 computer solution | Comment 5 of 17 |
`Part 1a:`
`list   10   dim Pr(16)   15   input Strt   20   Pr(1)=Strt   30   gosub *Addon(2)   40   end  200   *Addon(N)  210   X=nxtprm(Pr(N-1))  220   if N=3 and fnSq(Pr(1)+Pr(2)+X)=0 then X=nxtprm(X):goto 220  230   if N=5 and fnSq(Pr(4)+X)=0 then X=nxtprm(X):goto 220  240   if N=8 and fnSq(Pr(7)+X)=0 then X=nxtprm(X):goto 220  250   Pr(N)=X  260   if N=8 then  270     :for I=1 to 8:print Pr(I);:next:print:end  280    :else  290     :gosub *Addon(N+1)  300   return  500   fnSq(N)  510   Sr=int(sqrt(N)+0.5)  520   if Sr*Sr=N then return(1):else return(0)  530   returnOK`
` `

In the below runs, the "No gosub" response at the end has been omitted (resulting from ending the program with gosubs (calls) pending).

` `
`run? 2 2  3  11  13  23  29  31  113run? 3 3  5  17  19  557  563  569  587run? 5 5  7  13  17  19  23  29  71run? 7 7  11  31  37  107  109  113  211run? 11 11  13  97  101  223  227  229  347run? 13 13  17  19  23  41  43  47  53run? 17Break in 220run? 19 19  23  79  83  113  127  131  193run? 23 23  29  173  179  397  401  409  491 `

Note that a satisfactory solution starting with 17 was not being found, so the program was halted.

` `
`Part 1b:`
`list   10   dim Pr(16)   15   input Strt   20   Pr(1)=Strt   30   gosub *Addon(2)   40   end  200   *Addon(N)  210   X=nxtprm(Pr(N-1))  220   if N=3 and fnCu(Pr(1)+Pr(2)+X)=0 then X=nxtprm(X):goto 220  230   if N=5 and fnCu(Pr(4)+X)=0 then X=nxtprm(X):goto 220  240   if N=8 and fnCu(Pr(7)+X)=0 then X=nxtprm(X):goto 220  250   Pr(N)=X  260   if N=8 then  270     :for I=1 to 8:print Pr(I);:next:print:end  280    :else  290     :gosub *Addon(N+1)  300   return  500   fnCu(N)  510   Cr=int((N)^(1/3)+0.5)  520   if Cr*Cr*Cr=N then return(1):else return(0)  530   return`
`run? 2 2  3  59  61  1667  1669  1693  4139run? 3 3  5  19  23  41  43  47  953run? 5 5  7  113  127  1601  1607  1609  25391run? 7 7  11  107  109  1619  1621  1627  6373run? 11 11  13  101  103  113  127  131  1597run? 13 13  17  313  317  683  691  701  38603run? 17 17  19  89  97  2647  2657  2659  30109run? 19 19  23  83  89  127  131  137  863run? 23 23  29  73  79  137  139  149  1579run? 29 29  31  283  293  3803  3821  3823  4177No gosub`
`Part 2:`
`Changing line 20 of the cube version to`
`   20   Pr(1)=nxtprm(Strt)      Starting with the terminal numbers from part 1a:`
`run? 113 127  131  11909  11923  15077  15083  15091  58997run? 587 593  599  8069  8081  18919  18947  18959  27697run? 71 73  79  191  193  2551  2557  2579  3253run? 211 223  227  881  883  1861  1867  1871  11953run? 347 349  353  4211  4217  22783  22787  22807  152809run? 53 59  61  223  227  773  787  797  3299run? 193 197  199  1801  1811  4021  4027  4049  6599run? 491 499  503  3911  3917  9907  9923  9929  87407No gosub`
` `

The best in terms of the lowest final number seems to be:

` `
`  29  71  sq=100    23  17  19  sq=36    13    sq=25     7     5  followed by    2579  3253 cu=5832   2557 193  2551 cu=2744    191    cu=343     79     73`
`and also, for low numbers in general`
`  47  53  sq=100    43  23  41  sq=64    19    sq=49    17    13  followed by     797  3299 cu=4096    787 227   773 cu=1000    223    cu=343     61     59    `

 Posted by Charlie on 2010-01-06 15:44:13

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