Puzzle for December 27, 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 E and F to both sides of eq.2: C - E + E + F = E - F + E + F which becomes eq.2a) C + F = 2×E Add D and B to both sides of eq.5: A - D + D + B = D - B + D + B which becomes eq.5a) A + B = 2×D
Hint #2
Add D, E, A, F, and B to both sides of eq.6: C - D - E + B + D + E + A + F + B = D + E - A - F - B + D + E + A + F + B which becomes C + 2×B + A + F = 2×D + 2×E which is the same as eq.6a) C + F + 2×B + A = 2×D + 2×E
Hint #3
In eq.6a, replace C + F with 2×E (from eq.2a), and 2×D with A + B (from eq.5a): 2×E + 2×B + A = A + B + 2×E Subtract 2×E, B, and A from both sides of the above equation: 2×E + 2×B + A - 2×E - B - A = A + B + 2×E - 2×E - B - A which simplifies to B = 0
Hint #4
In eq.5a, substitute 0 for B: A + 0 = 2×D which makes A = 2×D
Hint #5
Substitute 2×D for A in eq.4: 2×D - F = F - D Add F and D to both sides of the above equation: 2×D - F + F + D = F - D + F + D which makes 3×D = 2×F Divide both sides by 2: 3×D ÷ 2 = 2×F ÷ 2 which makes 1½×D = F
Hint #6
Substitute 1½×D for F, and 0 for B in eq.3: 1½×D - E = 0 - 1½×D which becomes 1½×D - E = -1½×D Add E and 1½×D to both sides of the above equation: 1½×D - E + E + 1½×D = -1½×D + E + 1½×D which makes 3×D = E
Hint #7
Substitute 3×D for E, and 1½×D for F in eq.2: C - 3×D = 3×D - 1½×D which becomes C - 3×D = 1½×D Add 3×D to both sides of the equation above: C - 3×D + 3×D = 1½×D + 3×D which makes C = 4½×D
Solution
Substitute 2×D for A, 0 for B, 4½×D for C, 3×D for E, and 1½×D for F in eq.1: 2×D + 0 + 4½×D + D + 3×D + 1½×D = 24 which simplifies to 12×D = 24 Divide both sides of the above equation by 12: 12×D ÷ 12 = 24 ÷ 12 which means D = 2 making A = 2×D = 2 × 2 = 4 C = 4½×D = 4½ × 2 = 9 E = 3×D = 3 × 2 = 6 F = 1½×D = 1½ × 2 = 3 and ABCDEF = 409263