All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Six Digits = Triangle + Triangle (Posted on 2008-08-17) Difficulty: 3 of 5
There are three 6 digit numbers with the following properties applicable to each:

1. All digits are unique.
2. The first three digits ABC form a triangular number as do the latter three, DEF; both are multiples of 3.
3. The digital root/sum of the first triangular number is greater than that of the second.
4. Three consecutive digits form the difference of the triangular numbers, either being ascending or descending.

Identify the three 6 digit numbers.

See The Solution Submitted by brianjn    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution (spoiler) | Comment 1 of 3

There are 17 triangular numbers that are divisible by 3 and each consisting of 3 different digits:

105           6
120           3
153           9
210           3
231           6
276           6
351           9
378           9
435           3
465           6
528           6
561           3
630           9
741           3
780           6
861           6
903           3

shown here with the digital root of each.

Combining those as specified does produce three 6-digit numbers:

153276         9  6         123
465120         6  3         345
780435         6  3         345

where here the 6-digit number is shown with the digital roots of its two components and the difference between the two 3-digit triangular numbers.

DEFDBL A-Z
DIM tri(100), digRt(100)
triN = 1: addN = 1
DO
 addN = addN + 1
 triN = triN + addN
 IF triN > 100 AND triN < 1000 THEN
    tst$ = LTRIM$(STR$(triN))
    good = 1
    FOR i = 1 TO 2
      IF INSTR(i + 1, tst$, MID$(tst$, i, 1)) > 0 THEN good = 0: EXIT FOR
    NEXT
    IF triN MOD 3 <> 0 THEN good = 0
    IF good THEN
      triCt = triCt + 1: tri(triCt) = triN
      DO
        sum = 0
        FOR i = 1 TO LEN(tst$)
         sum = sum + VAL(MID$(tst$, i, 1))
        NEXT
        tst$ = LTRIM$(STR$(sum))
      LOOP UNTIL LEN(tst$) = 1
      digRt(triCt) = VAL(tst$)
      PRINT tri(triCt), digRt(triCt)
    END IF
 END IF
LOOP UNTIL triN > 999
PRINT triCt

FOR i = 1 TO triCt
 FOR j = 1 TO triCt
  IF i <> j THEN
    IF digRt(i) > digRt(j) THEN
      whole$ = LTRIM$(STR$(tri(i))) + LTRIM$(STR$(tri(j)))
      good = 1
      FOR ix = 1 TO 5
        IF INSTR(ix + 1, whole$, MID$(whole$, ix, 1)) > 0 THEN good = 0: EXIT FOR
      NEXT
      IF good THEN
        diff$ = LTRIM$(STR$(ABS(tri(i) - tri(j))))
        dig1 = VAL(MID$(diff$, 1, 1))
        dig2 = VAL(MID$(diff$, 2, 1))
        dig3 = VAL(MID$(diff$, 3, 1))
        IF dig2 - dig1 = dig3 - dig2 AND ABS(dig2 - dig1) = 1 THEN
          PRINT whole$, digRt(i); digRt(j), diff$
        END IF
      END IF
    END IF
  END IF
 NEXT
NEXT

 

Edited on August 17, 2008, 7:00 pm
  Posted by Charlie on 2008-08-17 18:58:46

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information