Puzzle for July 26, 2023 ( )
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
Add A and D to both sides of eq.3: B - A + A + D = A - D + A + D which becomes eq.3a) B + D = 2×A Add D and C to both sides of eq.4: C - D + D + C = B - C + D + C which becomes eq.4a) 2×C = B + D
Hint #2
In eq.4a, replace B + D with 2×A (from eq.3a): 2×C = 2×A Divide both sides of the above equation by 2: 2×C ÷ 2 = 2×A ÷ 2 which makes C = A
Hint #3
In eq.5, replace F with C + E (from eq.2), and B + D with 2×C (from eq.4a): E + C + E = 2×C which becomes 2×E + C = 2×C Subtract C from each side of the equation above: 2×E + C - C = 2×C - C which makes 2×E = C and also makes 2×E = C = A
Hint #4
Substitute 2×E for C in eq.2: F = 2×E + E which makes F = 3×E
Hint #5
Substitute 3×E for F in eq.5: E + 3×E = B + D which becomes 4×E = B + D Subtract D from each side of the above equation: 4×E - D = B + D - D which becomes eq.5a) 4×E - D = B
Hint #6
Substitute 4×E - D for B (from eq.5a) into eq.6: D - E = 4×E - D - D which becomes D - E = 4×E - 2×D Add E and 2×D to both sides of the equation above: D - E + E + 2×D = 4×E - 2×D + E + 2×D which makes 3×D = 5×E Divide both sides by 3: 3×D ÷ 3 = 5×E ÷ 3 which makes D = 1⅔×E
Hint #7
Substitute 1⅔×E for D in eq.3b: 4×E - 1⅔×E = B which makes 2⅓×E = B
Solution
Substitute 2×E for A and C, 2⅓×E for B, 1⅔×E for D, and 3×E for F in eq.1: 2×E + 2⅓×E + 2×E + 1⅔×E + E + 3×E = 36 which simplifies to 12×E = 36 Divide both sides of the above equation by 12: 12×E ÷ 12 = 36 ÷ 12 which means E = 3 making A = C = 2×E = 2 × 3 = 6 B = 2⅓×E = 2⅓ × 3 = 7 D = 1⅔×E = 1⅔ × 3 = 5 F = 3×E = 3 × 3 = 9 and ABCDEF = 676539