Puzzle for January 6, 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 E with B + D (from eq.2): C + B + D = A + D Subtract D from each side of the above equation: C + B + D – D = A + D – D which becomes C + B = A which is the same as eq.5a) B + C = A
Hint #2
In eq.3, replace B + C with A (from eq.5a): D = A + A which makes D = 2×A
Hint #3
In eq.6, replace E with B + D (from eq.2): D + F – B = A + B + B + D which becomes D + F – B = A + 2×B + D In the above equation, subtract D from both sides, and add B to both sides: D + F – B – D + B = A + 2×B + D – D + B which simplifies to eq.6a) F = A + 3×B
Hint #4
In eq.6a, replace A with B + C (from eq.5a): F = B + C + 3×B which becomes eq.6b) F = C + 4×B
Hint #5
In eq.4, substitute C + 4×B for F (from eq.6b), and 2×A for D: A + C + 4×B = C + 2×A Subtract A and C from both sides of the equation above: A + C + 4×B – A – C = C + 2×A – A – C which makes 4×B = A
Hint #6
Substitute 4×B for A in eq.5a: B + C = 4×B Subtract B from each side of the above equation: B + C – B = 4×B – B which makes C = 3×B
Hint #7
Substitute 4×B for A, and 3×B for C in eq.3: D = 4×B + B + 3×B which makes D = 8×B
Hint #8
Substitute 8×B for D in eq.2: E = B + 8×B which makes E = 9×B
Hint #9
Substitute 4×B for A in eq.6a: F = 4×B + 3×B which makes F = 7×B
Solution
Substitute 4×B for A, 3×B for C, 8×B for D, 9×B for E, and 7×B for F in eq.1: 4×B + B + 3×B + 8×B + 9×B + 7×B = 32 which simplifies to 32×B = 32 Divide both sides of the above equation by 32: 32×B ÷ 32 = 32 ÷ 32 which means B = 1 making A = 4×B = 4 × 1 = 4 C = 3×B = 3 × 1 = 3 D = 8×B = 8 × 1 = 8 E = 9×B = 9 × 1 = 9 F = 7×B = 7 × 1 = 7 and ABCDEF = 413897