Puzzle for August 17, 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.
Today's puzzle was sent to us by frequent contributor Judah S (age 15). Thank you, Judah!
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Hint #1
In eq.2, replace B with C + D (from eq.1): A + C = C + D + D which becomes A + C = C + 2×D Subtract C from each side of the equation above: A + C – C = C + 2×D – C which makes eq.2a) A = 2×D
Hint #2
eq.4 may be written as: D = (A + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (A + C + F) ÷ 3 which becomes eq.4a) 3×D = A + C + F
Hint #3
In eq.4a, substitute 2×D for A (from eq.2a): 3×D = 2×D + C + F Subtract 2×D from each side of the equation above: 3×D – 2×D = 2×D + C + F – 2×D which makes eq.4b) D = C + F
Hint #4
In eq.3, replace B with C + D (from eq.1): E + F = C + D – C which makes eq.3a) E + F = D
Hint #5
Substitute E + F for D (from eq.3a) into eq.4b: E + F = C + F Subtract F from each side of the above equation: E + F – F = C + F – F which makes E = C
Hint #6
In eq.5, substitute 2×D for A: 2×D ÷ D = B ÷ F which becomes 2 = B ÷ F Multiply both sides of the above equation by F: 2 × F = (B ÷ F) × F which becomes eq.5a) 2×F = B
Hint #7
Substitute C for E, and 2×F for B (from eq.5a) in eq.3: C + F = 2×F – C In the equation above, subtract F from each side, and add C to each side: C + F – F + C = 2×F – C – F + C which makes 2×C = F
Hint #8
Substitute (2×C) for F in eq.5a: 2×(2×C) = B which makes 4×C = B
Hint #9
Substitute 2×C for F in eq.4b: D = C + 2×C which makes D = 3×C
Hint #10
Substitute 3×C for D in eq.2a: A = 2×(3×C) which makes A = 6×C
Solution
Substitute 4×C for B, 2×C for F, 6×C for A, and C for E in eq.6: 4×C × 2×C = 6×C + C + C which becomes 8×C² = 8×C Since C ≠ 0 (3×C = D, and D ≠ 0 (from eq.5)), divide both sides of the above equation by 8×C: (8×C²) ÷ 8×C = (8×C) ÷ 8×C which means C = 1 making A = 6×C = 6 × 1 = 6 B = 4×C = 4 × 1 = 4 D = 3×C = 3 × 1 = 3 E = C = 1 F = 2×C = 2 × 1 = 2 and ABCDEF = 641312