Puzzle for April 10, 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.
Scratchpad
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Hint #1
In eq.3, replace A with C + D (from eq.2): C + D + E = F In eq.5, replace F with C + D + E: E + C + D + E = A + C + D Subtract both C and D from each side of the equation above: E + C + D + E - C - D = A + C + D - C - D which simplifies to 2×E = A
Hint #2
In eq.3, substitute 2×E for A: 2×E + E = F which makes eq.3a) 3×E = F
Hint #3
Add B to both sides of eq.6: B + C + B = F - B + B which becomes 2×B + C = F In eq.4, substitute 2×B + C for F: B + 2×B + C = A + C Subtract C from both sides: B + 2×B + C - C = A + C - C which simplifies to 3×B = A Substitute 2×E for A: 3×B = 2×E Divide both sides by 3: 3×B ÷ 3 = 2×E ÷ 3 which makes B = ⅔×E
Hint #4
Substitute ⅔×E for B, 3×E for F, and 2×E for A in eq.4: ⅔×E + 3×E = 2×E + C Subtract 2×E from both sides: ⅔×E + 3×E - 2×E = 2×E + C - 2×E which makes 1⅔×E = C
Hint #5
Substitute 1⅔×E for C, and 2×E for A in eq.2: 1⅔×E + D = 2×E Subtract 1⅔×E from each side: 1⅔×E + D - 1⅔×E = 2×E - 1⅔×E which makes D = ⅓×E
Solution
Substitute 2×E for A, ⅔×E for B, 1⅔×E for C, ⅓×E for D, and 3×E for F in eq.1: 2×E + ⅔×E + 1⅔×E + ⅓×E + E + 3×E = 26 which simplifies to 8⅔×E = 26 Divide both sides by 8⅔: 8⅔×E ÷ 8⅔ = 26 ÷ 8⅔ which means E = 1 making D = ⅓×E = ⅓ × 3 = 1 B = ⅔×E = ⅔ × 3 = 2 C = 1⅔×E = 1⅔ × 3 = 5 A = 2×E = 2 × 3 = 6 F = 3×E = 3 × 3 = 9 and ABCDEF = 625139