Problems & Puzzles:
Puzzles
Puzzle 63.- Another (3rd) Mike Keith's little
puzzle.
Take an n-digit integer and
construct all the possible expressions you can that use
the individual digits of the number combined with
parentheses and the operators +, -, *, and /. (Note: just
to be clear, because people sometimes get confused with
this kind of problem, NO other operators are allowed - no
exponentiation, no concatenation of digits, nothing.)
If you calculate the values of these
expressions, some will be non-integers, some will be
negative, etc, but just take those that are (positive)
prime numbers. Then find which range of consecutive
primes starting with 2 is included in the set. Call the
last prime in the range P(n).
Now...define a(n) as the largest
value of P(n) for any n-digit integer. What are the
values of a(n), and which primes produce them?
Here are the answers for the first
few n. Note that without loss of generality, we can
examine just n-digit integers whose digits in order are
non-decreasing. So that's the way I've listed them below.
n=2: 12, 13, or 36 generate 2...3;
so a(n) = 36
n=3: a(n) = 237 generates 2...23
n=4: a(n) = 3579 generates 2...71
n=5: a(n) = 45679 gives 2...317
n=6: a(n) = 567889 gives 2
1033
Here is n = 3 in detail:
2 = 7-3-2
3 = (7+2)/3
5 = (7+3)/2
7 = 7*(3-2)
11 = 2*7 - 3
13 = 2*3 + 7
17 = 2*7 + 3
19 = 7*3 - 2
23 = 7*3 + 2
a) Can you obtain the details for
n = 4, 5 & 6
b) Can you calculate a(n) for n= 7, 8 & 9?

Jim R. Howell sent (10/07/99) the
details for n = 4 (calculated 'by hand'):
2 = (7+9)/(3+5); 3 = (9-3)/(7-5); 5 =
5*(3+7-9); 7 = 7*(9-5-3); 11 = (7+3)/5+9; 13 =
5*3+7-9; 17 = 5*3+9-7; 19 = (9-5)*3+7; 23 = 7*5-3-9; 29 =
7*5+3-9; 31 = 5*3+7+9; 37 = (9-3)*5+7; 41 = 7*5+9-3; 43 =
(9+7)*3-5; 47 = 7*5+3+9; 53 = (9+3)*5-7; 59 = (7+3)*5+9;
61 = 9*7+3-5; 67 = (9+3)*5+7
71 = 9*7+5+3.
***
Giorgos Kalogeropoulos wrote on June 8, 2023:
Puzzle 63:
a) n=5,
2 = -(9+6*4-5*7) , 3 = 6-(5*4+7)/9 , 5 =
(9+5*6-4)/7 , 7 = 6-(7*5-4*9) , 11 = -4*9+7*6+5 , 13 = -4*5-9+7*6 ,
17 = 5*7-(6-4)*9 , 19 = 7*4-(6-5)*9 , 23 = 6*9-5*(7)+4 , 29 =
9+5*(7-6)*4 , 31 = -(7-6*4)+9+5 , 37 = 9*7-4*5-6 , 41 = 7+(6*9-4*5)
, 43 = 4*9-(5-6)*7 , 47 = 5+4/6*9*7 , 53 = 6*7-9+4*5 , 59 =
9*4+5*6-7 , 61 = 7+(5-4)*6*9 , 67 = 7*4-6+(5*9) , 71 = 9+4*5+7*6 ,
73 = 6*5+9*4+7 , 79 = 5*9+4*7+6 , 83 = -4+(7*6+5*9) , 89 = 5*4+6+9*7
, 97 = 5*6+4+9*7 , 101 = 4*9+(6+7)*5 , 103 = 7-(5-9)*6*4 , 107 =
9-(6-4*5)*7 , 109 = 9*6+(7+4)5 , 113 = 7*(5+6)+9*4 , 127 =
7*4+(6+5)*9 , 131 = -(4-6)*7*9+5 , 137 = 6-(9-5*7*4) , 139 =
(7+5*6)*4-9 , 149 = -7+(9+5*6)*4 , 151 = (9*4-7)*5+6 , 157 =
(9-4)*6*5+7 , 163 = 7-(6-5*9)*4 , 167 = -6+4*5*9-7 , 173 =
-9+(6+4*5)*7 , 179 = 4*5*9+(6-7) , 181 = -7*5+(6*9)*4 , 191 =
-(4-5*6)7+9 , 193 = 7+6+5*4*9 , 197 = -9+7*6*5-4 , 199 = (6*7-4)*5+9
, 211 = 4+(6*5-7)*9 , 223 = 4+9+(7*6*5) , 227 = (6+4*5)*9-7 , 229 =
-7+(5+9*6)*4 , 233 = 5+(9*7-6)*4 , 239 = -7+(4*9+5)*6 , 241 =
-(5-9*7*4+6) , 251 = +5-6+(4*7)*9 , 257 = 7+5*(9*6-4) , 263 =
5+(6+9*7*4) , 269 = (6*5+9)7-4 , 271 = (6+9*7)*4-5 , 277 =
4+(9*5-6)*7 , 281 = 4+7+(9*5*6) , 283 = (6*9+4)5-7 , 293 =
(9*5-4)*7+6 , 307 = 7+(6+9)5*4 , 311 = 9*(6+7*4)+5 , 313 =
7*5*9-(6-4) , 317 = 7*9*5-4+6.
n=6,
2 = -9*7+8*8+6-5 , 3 = 9*5-6*7*8/8 , 5 = -6+8*9-5-8*7
, 7 = (7*8-5)/9+8/6 , 11 = 5+6*9-7*8+8 , 13 = 6+(9/8*8-7+5) , 17 =
5+(8*9/8-7)*6 , 19 = 9*6-5/8*7*8 , 23 = 8-(5-6*9)/7+8 ,29 =
9*6-8*5+8+7 , 31 = (5+8)*6+9-7*8 , 37 = 7*9+6+8-5*8 , 41 =
7+8*9-6*5-8 ,43 = 6*8-8+5+7-9 , 47 = 5*6+8/8+9+7 , 53 =
5+6*9+8/8-7 , 59 = 9*5*(8-7)+6+8 , 61 = 8/6*9*8-7*5 , 67 =
6+8*8-7-5+9 , 71 = (9+5-6+7/8)*8 , 73 = 7*8+5+9/6*8 , 79 =
-7-8+(5*8+9*6) , 83 = 8-5*(6*8-9*7) , 89 = 8*6*9/8+7*5 , 97 = 8*9
7/6+8+5 , 101 = -(8-8*5-6)+7*9 , 103 = -7+5*6+9*8+8 , 107 =
9-5+6*(8+8)+7 , 109 = 9+(7+8)*8*5/6 , 113 = 5*8+7+9*8-6 , 127 =
5+8*9+8*7-6 , 131 = 9*8/6*8+7*5 , 137 = 7*8-(5-8-6)*9 , 139 =
-8/8+(9+5+6)*7 , 149 = 8+(8+5)*6+7*9 , 151 = 6*7+(5+8)*9-8 , 157 =
(7+8*9/6)8+5 , 163 = 8+8-(9-5*6)*7 , 167 = 6-7*(9+8-5*8) , 173 =
6*5+7+(8+9)*8 , 179 = 7*5-(6-8)*8*9 , 181 = 5-8*7*(6-9)+8 , 191 =
8*(7+6-8)*5-9 , 193 = (9/8+6*5-7)*8 , 197 = 7-8-(8-6*5)*9 , 199 =
-8*8+(6*5)*9-7 , 211 = 8+(8+6*5-9)*7 , 223 = -9+(5*8*7-8*6) , 227 =
-6+7*(5*8-8)+9 ,229 = (6*8+9)5-7*8 , 233 = 5*8*6+7*(8-9) , 239 =
-(9-7*8+8)*6+5 , 241 = -(6-7*5-9/8)*8 , 251 = 7*6*8-5*(9+8),257 =
-(8-9*5)*7+6-8 , 263 = 9*6*5-(8*7)/8 , 269 = 6-9+7*8*5-8 , 271 =
8*8-(7-6*5)*9 , 277 = 7+8/8*9*6*5 , 281 = (5*8+7)*6-9+8 , 283 =
-8-9*5+8*7*6 , 293 = 7+5*6*9+8+8 , 307 = 6*8+(5*9-8)*7 , 311 =
7+(5*9-6)*8-8 , 313 = (6-8+9*7)5+8 , 317 = -8+(8-6+7*9)5 , 331 =
5*(8-9)+7*8*6 , 337 = (8+8)+6+9*7*5 , 347 = (9+8*7-8)*6+5 , 349 =
9*(7*6-5)+8+8 , 353 = -6+8*5*9+7-8 , 359 = 7+8*(5+6*8-9) , 367 =
8+(5*8*9-7+6) , 373 = 8*(6*9*7-5)/8 , 379 = 8-5-(7-6*9)*8 , 383 =
8+(6*9*7)+5-8 , 389 = (6*9*8-8-5*7) , 397 = (6/8+9)5*8+7 , 401 =
6*8+8*9*5-7 , 409 = -7+8*(6+5*9)+8 , 419 = (7+6*9-8)*8-5 , 421 =
(5*9+6)*7+8*8 , 431 = (5+8*7-6)*8-9 , 433 = 6+(8+5*9+8)*7 , 439 =
8*8*7-(6-5)*9 , 443 = 5-(7-8*9-8)*6 , 449 = 6+(8*9-8)*7-5 , 457 =
9+(6-5)*8*8*7 , 461 = -7*5+(9*6+8)*8 , 463 = (8+6 9-5)*8+7 , 467 =
8*(5*7/8+6*9) , 479 = 8*6*9+7+8*5 , 487 = 9*5+(8*7*8-6) , 491 =
(5+9*8)*7-6*8 , 499 = 6+(9*5+8*8*7) , 503 = 5*(7+6)8-9-8 , 509 =
7+(9+8)*6*5-8 , 521 = (6+8*7-5)*9+8 , 523 = 6+5+8+9*7*8 , 541 =
(6/8+9)8*7-5 , 547 = 6*8+8*9*7-5 , 557 = 8*7*9+(8*6+5), 563 =
8+(9*7+6*8)*5 , 569 = (8-6)5*7*8+9 , 571 = 8*8*9+5/(6-7) , 577 =
9+(7+5)*6*8-8 , 587 = -8/8+(5+9)*7*6 , 593 = 6-8+5*7*(9+8) , 599 =
5*6-7+9*8*8 , 601 = 7+(8+8-5)*9*6 , 607 = -9+(5*8+8*6)*7 , 613 =
(7+8*9*8)+5*6 , 617 = -8+(6+5)8*7+9 , 619 = (5+8)*(6*9-7)+8 , 631 =
5-6+8*(9*8+7) , 641 = (7+6)5+8*9*8 , 643 = (6*8+9 5)7-8 , 647 =
7+8*(9+6-5)8 , 653 = 5+(7+8-6)8*9 , 659 = 8*(6+9*8)+7*5 , 661 =
7+6*(8*8+5*9) , 673 = 9+(7*5+8*6)*8 , 677 = 5+(8*8*7*9)/6 , 683 =
-7-5(6-(8+8)9) , 691 = 6-5(7-(8+8)*9) , 701 = 6*9*(5+8)+7-8 , 709 =
8+(5+6)*9*7+8 , 719 = 9*(8+8)*5+6-7 , 727 = 7+9*(8-6)8*5 , 733 =
7+6+(8+8)*9*5 , 739 = 7*(6+8)8-5*9 , 743 = 6+8*7*(8+5)+9 , 751 =
7+(9*5+8*6)*8 , 757 = (5+6)*9*7+8*8 , 761 = 5*(8+6+8)7-9 , 769 =
7*(8*8+9*5)+6 , 773 = 5+6*(7*8+9*8) , 787 = -9*5+8*(6+7)*8 , 797 =
-5-8+(7+8)*9*6 , 809 = (9*8+6*5)*8-7 , 811 = (8(9+7)+8)6-5 , 821 =
8-6+(5+8)9*7 , 823 = 8+5+(8+7)*9*6 , 827 = (6+9)7*8-5-8 , 829 =
7*(6+8)*8+9*5 , 839 = -7+(8*8+6*5)*9 , 853 = 7+(6*5+8*8)*9 , 857 =
(8*(5+6)+8)9-7 , 859 = -5-9*(8-(7+6)*8) , 863 = 5+(9(7+8)+8)*6 , 877
= 8*8*(6+7)+5*9 , 881 = 6+(9+8+8)*7*5 , 883 = (9*7+6 8)*8-5 , 887 =
(5+(8+8)9)*6-7 , 907 = -5+8*(6*7+8*9) , 911 = (7+(8+8)9)6+5 , 919 =
-8+(7(8+6)+5)*9 , 929 = (8+5)*(8+7*9)+6 , 937 = 9*(7+8+6)*5-8 , 941
= (7*8+8*6)*9+5 , 947 = -5-8(7-(8+6)*9) , 953 = -7+8*8*5*(9-6) , 967
= 7-(6-9)5*8*8 , 971 = -5-8(6-(9+7)8) , 977 = -7+8*(6+(5+8)*9) , 983
= -7+(6+8+8)5*9 , 991 = 7+(9(5+8)+6)*8 , 997 = (8+8)*9*7-6-5 , 1009
= 6+(8+8)9*7-5 , 1013 = (8-6)8*9*7+5 , 1019 = 5+6+(8+8)*7*9 , 1021 =
8(7+(6+9)8)+5 , 1031 = 5-9(6-(7+8)*8) , 1033 = (7+8*9)*(8+5)+6
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