Puzzle for February 27, 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.6, replace B + E with A + C - E (from eq.5): A + D = A + C - E + F Add E to each side, and subtract A from each side of the above equation: A + D + (E - A) = A + C - E + F + (E - A) which becomes D + E = C + F In eq.4, replace D + E with C + F: C + F = B + C Subtract C from both sides: C + F - C = B + C - C which makes F = B
Hint #2
In eq.3, replace F with B: B + D = A + B Subtract B from each side of the above equation: B + D - B = A + B - B which makes D = A
Hint #3
In eq.2, replace F with B: C + E = B In eq.4, substitute C + E for B: D + E = C + E + C which becomes D + E = 2×C + E Subtract E from each side of the above equation: D + E - E = 2×C + E - E which makes D = 2×C which also means A = D = 2×C
Hint #4
Substitute 2×C for A, and F for B in eq.5: 2×C + C - E = F + E which becomes 3×C - E = F + E Substitute C + E for F (from eq.2) in the above equation: 3×C - E = C + E + E which becomes 3×C - E = C + 2×E Add (E - C) to both sides: 3×C - E + (E - C) = C + 2×E + (E - C) which simplifies to 2×C = 3×E Divide both sides by 3: 2×C ÷ 3 = 3×E ÷ 3 which makes ⅔×C = E
Hint #5
Substitute ⅔×C for E in eq.2: C + ⅔×C = F which makes 1⅔×C = F which also makes B = F = 1⅔×C
Solution
Substitute 2×C for A and D, 1⅔×C for B and F, and ⅔×C for E in eq.1: 2×C + 1⅔×C + C + 2×C + ⅔×C + 1⅔×C = 27 which simplifies to 9×C = 27 Divide each side by 9: 9×C ÷ 9 = 27 ÷ 9 which makes C = 3 making A = D = 2×C = 2 × 3 = 6 B = F = 1⅔×C = 1⅔ × 3 = 5 E = ⅔×C = ⅔ × 3 = 2 and ABCDEF = 653625