Puzzle for October 16, 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 D to both sides of eq.5: C + D + D = E - D + D which becomes eq.5a) C + 2×D = E In eq.3, replace E with C + 2×D (from eq.5a): C + 2×D = B + C Subtract C from both sides of eq.3a: C + 2×D - C = B + C - C which makes eq.3a) 2×D = B
Hint #2
In eq.6, replace B with 2×D (from eq.3a): 2×D - C = C + D In the above equation, add C to both sides, and subtract D from both sides: 2×D - C + C - D = C + D + C - D which makes D = 2×C
Hint #3
In eq.3a, substitute (2×C) for D: 2×(2×C) = B which makes 4×C = B
Hint #4
Substitute 4×C for B in eq.3: E = 4×C + C which makes E = 5×C
Hint #5
Substitute 4×C for B, and 5×C for E in eq.3: F = 4×C + 5×C which makes F = 9×C
Hint #6
Substitute 2×C for D, and 5×C for E in eq.4: A = C + 2×C + 5×C which makes A = 8×C
Solution
Substitute 8×C for A, 4×C for B, 2×C for D, 5×C for E, and 9×C for F in eq.1: 8×C + 4×C + C + 2×C + 5×C + 9×C = 29 which simplifies to 29×C = 29 Divide both sides of the above equation by 29: 29×C ÷ 29 = 29 ÷ 29 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 4×C = 4 × 1 = 4 D = 2×C = 2 × 1 = 2 E = 5×C = 5 × 1 = 5 F = 9×C = 9 × 1 = 9 and ABCDEF = 841259