Puzzle for April 19, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* BC, CD, and EF are 2-digit numbers (not B×C, C×D, or E×F).
** ABC and BCD are 3-digit numbers (not A×B×C or B×C×D).
Scratchpad
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Hint #1
In eq.4, substitute (B + E) for D (from eq.3): B + C - (B + E) + E = F which is equivalent to B + C - B - E + E = F which simplifies to C = F
Hint #2
eq.5 may be written as: A + 10×B + C + 10×E + F = 10×C + D Subtract C from both sides of the above equation: A + 10×B + C + 10×E + F - C = 10×C + D - C which becomes eq.5a) A + 10×B + 10×E + F = 9×C + D
Hint #3
eq.6 may be written as: 100×B + 10×C + D - (10×E + F) = A + 100×A + 10×B + C which is the same as 100×B + 10×C + D - 10×E - F = A + 100×A + 10×B + C Subtract both 10×B and C from each side of the equation above: 100×B + 10×C + D - 10×E - F - 10×B - C = A + 100×A + 10×B + C - 10×B - C which becomes eq.6a) 90×B + 9×C + D - 10×E - F = 101×A
Hint #4
In eq.6a, substitute A + 10×B + 10×E + F for 9×C + D (from eq.5a): 90×B + A + 10×B + 10×E + F - 10×E - F = 101×A which simplifies to 100×B + A = 101×A Subtract A from both sides: 100×B + A - A = 101×A - A which means 100×B = 100×A Divide each side by 100: 100×B ÷ 100 = 100×A ÷ 100 which makes B = A
Hint #5
Substitute A for B in eq.2: C = A + A which makes C = 2×A which also makes F = C = 2×A
Hint #6
Substitute A for B in eq.3: eq.3a) D = A + E Substitute A for B, (2×A) for C and F, and A + E for D (from eq.3a) in eq.5a: A + 10×A + 10×E + 2×A = 9×(2×A) + A + E which becomes 13×A + 10×E = 18×A + A + E Subtract both 13×A and E from each side: 13×A + 10×E - 13×A - E = 18×A + A + E - 13×A - E which simplifies to 9×E = 6×A Divide both sides by 9: 9×E ÷ 9 = 6×A ÷ 9 which means E = ⅔×A
Hint #7
Substitute ⅔×A for E in eq.3a: D = A + ⅔×A which makes D = 1⅔×A
Solution
Substitute A for B, 2×A for C and F, 1⅔×A for D, and ⅔×A for E in eq.1: A + A + 2×A + 1⅔×A + ⅔×A + 2×A = 25 which becomes 8⅓×A = 25 Divide both sides by 8⅓: 8⅓×A ÷ 8⅓ = 25 ÷ 8⅓ which makes A = 3 making B = A = 3 C = F = 2×A = 2 × 3 = 6 D = 1⅔×A = 1⅔ × 3 = 5 E = ⅔×A = ⅔ × 3 = 2 and ABCDEF = 336526