Puzzle for January 13, 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
Add E to both sides of eq.3: C + E + E = A – E + F + E which becomes eq.3a) C + 2×E = A + F In eq.3a, replace A + F with B + E (from eq.4): C + 2×E = B + E Subtract E from both sides of the above equation: C + 2×E – E = B + E – E which becomes eq.3b) C + E = B
Hint #2
In eq.2, replace B with C + E (from eq.3b): C + E – D = C + D In the above equation, subtract C from both sides, and add D to both sides: C + E – D – C + D = C + D – C + D which makes E = 2×D
Hint #3
In eq.6, substitute 2×D for E: D + F = 2×D Subtract D from both sides of the equation above: D + F – D = 2×D – D which makes F = D
Hint #4
Substitute (2×D) for E, and D for F in eq.3a: C + 2×(2×D) = A + D which becomes C + 4×D = A + D Subtract D from each side of the equation above: C + 4×D – D = A + D – D which becomes eq.3c) C + 3×D = A
Hint #5
Substitute 2×D for E in eq.3b: eq.3d) C + 2×D = B
Hint #6
Substitute C + 2×D for B (from eq.3d), C + 3×D for A (from eq.3c), 2×D for E, and D for F in eq.5: C + 2×D + C = C + 3×D + D + 2×D + D which becomes 2×C + 2×D = C + 7×D Subtract 2×D and C from both sides of the equation above: 2×C + 2×D – 2×D – C = C + 7×D – 2×D – C which simplifies to C = 5×D
Hint #7
Substitute 5×D for C in eq.3c: 5×D + 3×D = A which makes 8×D = A
Hint #8
Substitute 5×D for C in eq.3d: 5×D + 2×D = B which makes 7×D = B
Solution
Substitute 8×D for A, 7×D for B, 5×D for C, 2×D for E, and D for F in eq.1: 8×D + 7×D + 5×D + D + 2×D + D = 24 which simplifies to 24×D = 24 Divide both sides of the equation above by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = 8×D = 8 × 1 = 8 B = 7×D = 7 × 1 = 7 C = 5×D = 5 × 1 = 5 E = 2×D = 2 × 1 = 2 F = D = 1 and ABCDEF = 875121