Please restore the 5 by 5 grid containing 10 ternary numbers with no leading zeroes.
The crossword-like definitions follow:
Across:
I. a cube
II. twice a permutable (in base 10) prime
III. divisible by eleven
IV. the number of trees with 10 vertices
V. a square
Down:
1. repdigit in base 12
2. a factorion in base 10
3. a semiprime
4. a square of a prime
5. its cube in base 10 uses only 3 distinct digits
Rem: I have built it bottoms-up. Hope there is only one solution...
DECLARE SUB factor (num%, s$)
DECLARE FUNCTION fact# (x#)
DECLARE SUB permute (a$)
DECLARE FUNCTION bs12$ (x#)
DECLARE FUNCTION tern$ (x#)
DEFDBL A-Z
DIM prime(164) AS STRING
DATA 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43
DATA 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89
DATA 97 , 101 , 103 , 107 , 109 , 113 , 127 , 131 , 17 , 139
DATA 149 , 151 , 157 , 163 , 167 , 173 , 179 , 181 , 191 , 193
DATA 197 , 199 , 211 , 223 , 227 , 229 , 233 , 239 , 241 , 251
DATA 257 , 263 , 269 , 271 , 277 , 281 , 283 , 293 , 307 , 311
DATA 313 , 317 , 331 , 337 , 347 , 349 , 353 , 359 , 367 , 373
DATA 379 , 383 , 389 , 397 , 401 , 409 , 419 , 421 , 431 , 433
DATA 439 , 443 , 449 , 457 , 461 , 463 , 467 , 479 , 487 , 491
DATA 499 , 503 , 509 , 521 , 523 , 541 , 547 , 557 , 563 , 569
DATA 571 , 577 , 587 , 593 , 599 , 601 , 607 , 613 , 617 , 619
DATA 631 , 641 , 643 , 647 , 653 , 659 , 661 , 673 , 677 , 683
DATA 691 , 701 , 709 , 719 , 727 , 733 , 739 , 743 , 751 , 757
DATA 761 , 769 , 773 , 787 , 797 , 809 , 811 , 821 , 823 , 827
DATA 829 , 839 , 853 , 857 , 859 , 863 , 877 , 881 , 883 , 887
DATA 907 , 911 , 919 , 929 , 937 , 941 , 947 , 953 , 967 , 971
DATA 977 , 983 , 991 , 997
FOR i = 1 TO 164: READ prime(i): NEXT
CLS
low = 3 ^ 4: high = 3 ^ 5 - 1: PRINT low, high
PRINT tern$(low), tern$(high)
PRINT : PRINT "across"
OPEN "terntran.txt" FOR OUTPUT AS #2
PRINT "cube"
FOR n = low TO high
cr = INT(n ^ (1 / 3) + .5)
IF cr * cr * cr = n THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "twice perm prime"
FOR n = low TO high
IF n MOD 2 = 0 THEN
FOR i = 1 TO 164
IF VAL(prime(i)) = n / 2 THEN
good = 0
a$ = prime(i): h$ = a$
DO
FOR j = 1 TO 164
IF prime(j) = a$ AND VAL(a$) <> n / 2 THEN good = 1: EXIT FOR
NEXT
IF good THEN EXIT DO
permute a$
LOOP UNTIL a$ = h$
IF good THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
END IF
NEXT
END IF
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "div by 11"
FOR n = low TO high
IF n MOD 11 = 0 THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "trees"
n = 106
PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "square"
FOR n = low TO high
sr = INT(SQR(n) + .5)
IF sr * sr = n THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "--------------"
PRINT #2, "--------------"
PRINT "down"
PRINT "repdigit base 12"
FOR n = low TO high
x$ = bs12$(n)
good = 1
FOR i = 2 TO LEN(x$)
IF MID$(x$, i, 1) <> LEFT$(x$, 1) THEN good = 0: EXIT FOR
NEXT
IF good THEN PRINT n; bs12$(n); " "; tern$(n); " "; : PRINT #2, tern$(n); " ";
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "factorion base 10"
FOR n = low TO high
tot = 0
ns$ = LTRIM$(STR$(n))
FOR i = 1 TO LEN(ns$)
tot = tot + fact(VAL(MID$(ns$, i, 1)))
NEXT
IF tot = n THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "semiprime"
FOR n = low TO high
factor INT(n), f$
ix = INSTR(f$, " ")
IF ix > 0 THEN
ix = INSTR(ix + 1, f$, " ")
IF ix = 0 THEN
PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
END IF
END IF
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
PRINT "square of prime"
FOR n = low TO high
factor INT(n), f$
ix = INSTR(f$, " ")
IF ix > 0 THEN
ix2 = INSTR(ix + 1, f$, " ")
IF ix2 = 0 THEN
s1$ = LEFT$(f$, ix - 1)
s2$ = MID$(f$, ix + 1)
IF s1$ = s2$ THEN PRINT n; tern$(n); " "; : PRINT #2, tern$(n); " ";
END IF
END IF
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
FOR n = low TO high
n3 = n * n * n
n3s$ = LTRIM$(STR$(n3))
dct = 1
FOR i = 2 TO LEN(n3s$)
IF INSTR(n3s$, MID$(n3s$, i, 1)) = i THEN dct = dct + 1
NEXT
IF dct <= 3 THEN
PRINT n; n3; dct; tern$(n); " "; : PRINT #2, tern$(n); " ";
END IF
NEXT: PRINT : PRINT : PRINT #2, : PRINT #2,
FUNCTION bs12$ (x)
s$ = ""
n = x
WHILE n > 0
d = n MOD 12: n = n \ 12
s$ = LTRIM$(MID$("0123456789ab", d + 1, 1)) + s$
WEND
bs12$ = s$
END FUNCTION
FUNCTION fact (x)
f = 1
FOR i = 2 TO x
f = f * i
NEXT
fact = f
END FUNCTION
DEFINT A-Z
SUB factor (num, s$)
s$ = "": n = ABS(num): IF n > 0 THEN limit = SQR(n): ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
dv = 2: GOSUB DivideIt
dv = 3: GOSUB DivideIt
dv = 5: GOSUB DivideIt
dv = 7
DO UNTIL dv > limit
GOSUB DivideIt: dv = dv + 4 '11
GOSUB DivideIt: dv = dv + 2 '13
GOSUB DivideIt: dv = dv + 4 '17
GOSUB DivideIt: dv = dv + 2 '19
GOSUB DivideIt: dv = dv + 4 '23
GOSUB DivideIt: dv = dv + 6 '29
GOSUB DivideIt: dv = dv + 2 '31
GOSUB DivideIt: dv = dv + 6 '37
IF INKEY$ = CHR$(27) THEN s$ = CHR$(27): EXIT SUB
LOOP
IF n > 1 THEN s$ = s$ + STR$(n)
s$ = LTRIM$(s$)
EXIT SUB
DivideIt:
DO
q = INT(n / dv)
IF q * dv = n AND n > 0 THEN
n = q: s$ = s$ + STR$(dv): IF n > 0 THEN limit = SQR(n): ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
ELSE
EXIT DO
END IF
LOOP
RETURN
END SUB
DEFDBL A-Z
FUNCTION tern$ (x)
s$ = ""
n = x
WHILE n > 0
d = n MOD 3: n = n \ 3
s$ = LTRIM$(STR$(d)) + s$
WEND
tern$ = s$
END FUNCTION
produces the below possibilities:
Unless otherwise noted, the groups consist of the decimal value followed by the ternary value.
across
cube
125 11122 216 22000
twice perm prime
142 12021 146 12102 158 12212 194 21012 214 21221 226 22101
div by 11
88 10021 99 10200 110 11002 121 11111 132 11220 143 12022 154 12201 165
20010 176 20112 187 20221 198 21100 209 21202 220 22011 231 22120 242
22222
trees
106 10221
square
81 10000 100 10201 121 11111 144 12100 169 20021 196 21021 225 22100
--------------
down
repdigit base 12
(groups are base-10, base-12, base-3)
91 77 10101 104 88 10212 117 99 11100 130 aa 11211 143 bb 12022 157 111
12211
factorion base 10
145 12101
semiprime
82 10001 85 10011 86 10012 87 10020 91 10101 93 10110 94 10111 95 10112
106 10221 111 11010 115 11021 118 11101 119 11102 121 11111 122 11112
123 11120 129 11210 133 11221 134 11222 141 12020 142 12021 143 12022
145 12101 146 12102 155 12202 158 12212 159 12220 161 12222 166 20011
169 20021 177 20120 178 20121 183 20210 185 20212 187 20221 194 21012
201 21110 202 21111 203 21112 205 21121 206 21122 209 21202 213 21220
214 21221 215 21222 217 22001 218 22002 219 22010 221 22012 226 22101
235 22201 237 22210
square of prime
121 11111 169 20021
cube in base 10 uses 3 distinct digits or fewer:
(groups consist of decimal value, cube, number of dist. digits, ternary rep):
92 778688 3 10102 100 1000000 2 10201 101 1030301 3 10202 110
1331000 3 11002 173 5177717 3 20102 192 7077888 3 21010 200 8000000
2 21102 211 9393931 3 21211
Summary of the ternary values:
across
11122 22000
12021 12102 12212 21012 21221 22101
10021 10200 11002 11111 11220 12022 12201 20010 20112 20221 21100 21202 22011 22120 22222
10221
10000 10201 11111 12100 20021 21021 22100
--------------
down
10101 10212 11100 11211 12022 12211
12101
10001 10011 10012 10020 10101 10110 10111 10112 10221 11010 11021 11101 11102 11111 11112 11120 11210 11221 11222 12020 12021 12022 12101 12102 12202 12212 12220 12222 20011 20021 20120 20121 20210 20212 20221 21012 21110 21111 21112 21121 21122 21202 21220 21221 21222 22001 22002 22010 22012 22101 22201 22210
11111 20021
10102 10201 10202 11002 20102 21010 21102 21211
Obviously the fourth row across is 10221 and then the fourth column down becomes 20021. Similar fittings are made until the square is complete:
1 1 1 2 2 cube = 125 decimal
2 2 1 0 1 twice permutable prime = 226 = 2*113
2 1 2 0 2 div by 11: = 209
1 0 2 2 1 trees w/10 vert. = 106 decimal
1 1 1 1 1 square = 121 decimal
down
12211 repdigit base 12: base-10 157; base-12 111
12101 factorion base 10: 145 = 1! + 4! + 5!
11221 semiprime = 133 decimal = 7*19
20021 square of prime = 169 decimal = 13^2
21211 cube contains only 3 distinct decimal digits: decimal 211^3 = 9393931
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Posted by Charlie
on 2014-03-28 14:26:02 |
computer solution
