Puzzle for February 7, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
Help Area
Hint #1
In eq.5, replace E + F with C + D (from eq.3): A + D = B + C + D Subtract D from both sides of the above equation: A + D - D = B + C + D - D which becomes A = B + C Replace B + C with D (from eq.2): A = D
Hint #2
In eq.4, replace D with A: A + E = A + F Subtract A from both sides of the above equation: A + E - A = A + F - A which simplifies to E = F
Hint #3
Substitute E for F, and A for D in eq.3: E + E = C + A which becomes 2×E = C + A Subtract A from both sides of the above equation: 2×E - A = C + A - A which becomes eq.3a) 2×E - A = C
Hint #4
Substitute A for D, and E for F in eq.5: A + A = B + E + E which becomes 2×A = B + 2×E Subtract 2×E from both sides of the above equation: 2×A - 2×E = B + 2×E - 2×E which makes eq.5a) 2×A - 2×E = B
Hint #5
eq.6 may be written as: 10×A + B = C + D + 10×E + F Substitute A for D, and E for F in the above equation: 10×A + B = C + A + 10×E + E Subtract A from each side: 10×A + B - A = C + A + 10×E + E - A which becomes eq.6a) 9×A + B = C + 11×E
Hint #6
Substitute 2×A - 2×E for B (from eq.5a), and 2×E - A for C (from eq.3a) in eq.6a: 9×A + 2×A - 2×E = 2×E - A + 11×E which becomes 11×A - 2×E = 13×E - A Add A + 2×E to both sides of the above equation: 11×A - 2×E + A + 2×E = 13×E - A + A + 2×E which simplifies to 12×A = 15×E Divide both sides by 12: 12×A ÷ 12 = 15×E ÷ 12 A = 1¼×E
Hint #7
In eq.3a, replace A with 1¼×E: 2×E - 1¼×E = C which makes ¾×E = C
Hint #8
In eq.5a, replace A with (1¼×E): 2×(1¼×E) - 2×E = B which becomes 2½×E - 2×E = B which means ½×E = B
Solution
Substitute 1¼×E for A and D, ½×E for B, ¾×E for C, and E for F in eq.1: 1¼×E + ½×E + ¾×E + 1¼×E + E + E = 23 which simplifies to 5¾×E = 23 Divide both sides by 5¾: 5¾×E ÷ 5¾ = 23 ÷ 5¾ which becomes E = 4 making A = D = 1¼×E = 1¼ × 4 = 5 B = ½×E = ½ × 4 = 2 C = ¾×E = ¾ × 4 = 3 F = E = 4 and ABCDEF = 523544