Each of the capital letters in bold represents a different digit from 0 to 9. None of the numbers can contain a leading zero.

1. TWO is a prime number, and:
2. THREE is a triangular number, and:
3. FOUR is a perfect square.

Determine the numbers represented by:

(i) THREE   (ii) FOUR

*** *** Adapted from Enigma No. 1341 by Richard England appearing in "New Scientist" on May 21, 2005.

DECLARE SUB permute (a$)
CLS

DATA 2,3,5,7,11,13,17,19,23,29,31,37,41
DATA 43,47,53,59,61,67,71,73,79,83,89,97
DATA 101,103,107,109,113,127,131,137,139,149
DATA 151,157,163,167,173,179,181,191,193,197
DATA 199,211,223,227,229,233,239,241,251,257
DATA 263,269,271,277,281,283,293,307,311,313
DATA 317,331,337,347,349,353,359,367,373,379,383
DATA 389,397,401,409,419,421,431,433,439,443,449,457
DATA 461,463,467,479,487,491,499,503,509,521,523
DATA 541,547,557,563,569,571,577,587,593,599,601,607
DATA 613,617,619,631,641,643,647,653,659,661,673,677,683
DATA 691,701,709,719,727,733,739,743,751,757,761,769,773
DATA 787,797,809,811,821,823,827,829,839,853,857,859,863
DATA 877,881,883,887,907,911,919,929,937,941,947,953,967
DATA 971,977,983,991,997,1009

a$ = "1234567890": h$ = a$
DO
    IF INSTR(a$, "0") > 2 THEN
        t = VAL(MID$(a$, 1, 1))
        f = VAL(MID$(a$, 2, 1))
        w = VAL(MID$(a$, 3, 1))
        o = VAL(MID$(a$, 4, 1))
        h = VAL(MID$(a$, 5, 1))
        r = VAL(MID$(a$, 6, 1))
        e = VAL(MID$(a$, 7, 1))
        u = VAL(MID$(a$, 8, 1))
        two = t * 100 + w * 10 + o
        three = 10000 * t + 1000 * h + 100 * r + 11 * e
        four = 1000 * f + 100 * o + 10 * u + r
        IF INKEY$ = CHR$(27) THEN STOP
        IF isTri(three) THEN
            sr = INT(SQR(four) + .5)
            IF sr * sr = four THEN
                RESTORE
                found = 0
                DO
                    READ pr: IF pr > 999 THEN EXIT DO
                    IF pr = two THEN found = 1: EXIT DO
                LOOP
                IF found THEN PRINT two; three; four
            END IF
        END IF
    END IF
    permute a$
LOOP UNTIL a$ = h$

SUB permute (a$)

x$ = ""
FOR i = LEN(a$) TO 1 STEP -1
    l$ = x$
    x$ = MID$(a$, i, 1)
    IF x$ < l$ THEN EXIT FOR
NEXT

IF i = 0 THEN
    FOR j = 1 TO LEN(a$) \ 2
        x$ = MID$(a$, j, 1)
        MID$(a$, j, 1) = MID$(a$, LEN(a$) - j + 1, 1)
        MID$(a$, LEN(a$) - j + 1, 1) = x$
    NEXT
ELSE
    FOR j = LEN(a$) TO i + 1 STEP -1
        IF MID$(a$, j, 1) > x$ THEN EXIT FOR
    NEXT
    MID$(a$, i, 1) = MID$(a$, j, 1)
    MID$(a$, j, 1) = x$
    FOR j = 1 TO (LEN(a$) - i) \ 2
        x$ = MID$(a$, i + j, 1)
        MID$(a$, i + j, 1) = MID$(a$, LEN(a$) - j + 1, 1)
        MID$(a$, LEN(a$) - j + 1, 1) = x$
    NEXT
END IF
END SUB

FUNCTION isTri (t)
n = INT(SQR(t * 2))
np = n + 1
IF n * np = 2 * t THEN isTri = n: ELSE isTri = 0
END FUNCTION

finds

137  15400  6724
197  15400  6724

Posted by Charlie on 2014-04-08 16:37:41