Puzzle for February 9, 2022 ( )
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.5, replace C with D + E (from eq.2): B + D – F = D + E – B – E which becomes B + D – F = D – B In the above equation, subtract D from both sides, and add F and B to both sides: B + D – F – D + F + B = D – B – D + F + B which simplifies to eq.5a) 2×B = F
Hint #2
In eq.4, replace F with 2×B (from eq.5a): C + 2×B – D = A + B + E In the equation above, add D to both sides, and subtract B from both sides: C + 2×B – D + D – B = A + B + E + D – B which becomes C + B = A + E + D which may be written as eq.4a) B + C = A + D + E
Hint #3
In eq.4a, substitute C for D + E (from eq.2): B + C = A + C Subtract C from each side of the equation above: B + C – C = A + C – C which makes B = A
Hint #4
Substitute 2×B for F (from eq.5a), B for A, and (D + E) for C (from eq.2) in eq.6: 2×B – E = B + B – (D + E) – D + E which becomes 2×B – E = 2×B – D – E – D + E which becomes 2×B – E = 2×B – 2×D Subtract 2×B from each side of the above equation: 2×B – E – 2×B = 2×B – 2×D – 2×B which becomes –E = –2×D Multiply both sides by (–1): –E × (–1) = –2×D × (–1) which becomes E = 2×D
Hint #5
Substitute 2×D for E in eq.2: D + 2×D = C which makes 3×D = C
Hint #6
Substitute 2×D for E, and B for A in eq.3: B – D = 2×D – B Add B and D to both sides of the above equation: B – D + B + D = 2×D – B + B + D which becomes 2×B = 3×D Divide both sides by 2: 2×B ÷ 2 = 3×D ÷ 2 which makes B = 1½×D and also makes A = B = 1½×D
Hint #7
Substitute (1½×D) for B in eq.5a: 2×(1½D) = F which makes 3×D = F
Solution
Substitute 1½×D for A and B, 3×D for C and F, and 2×D for E in eq.1: 1½×D + 1½×D + 3×D + D + 2×D + 3×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 = B = 1½×D = 1½ × 2 = 3 C = F = 3×D = 3 × 2 = 6 E = 2×D = 2 × 2 = 4 and ABCDEF = 336246