Puzzle for July 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.
Once again, we thank Judah S (age 15) for sending us a fun and interesting puzzle. Thank you, Judah!
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Hint #1
eq.2 may be written as: F = A + E + B In the above equation, replace A + E with B (from eq.1): F = B + B which makes F = 2×B
Hint #2
In eq.4, replace F with 2×B: E = 2×B ÷ B which makes E = 2
Hint #3
In eq.1, substitute 2 for E: B = A + 2 Subtract 2 from each side of the above equation: B – 2 = A + 2 – 2 which becomes eq.1a) B – 2 = A
Hint #4
In eq.6, substitute B – 2 for A (from eq.1a), 2×B for F, and 2 for E: B – 2 + 2×B = C + (2 × 2) which becomes 3×B – 2 = C + 4 Subtract 4 from each side of the equation above: 3×B – 2 – 4 = C + 4 – 4 which becomes eq.6a) 3×B – 6 = C
Hint #5
eq.3 may be written as: D = (A + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × D = 3 × (A + E + F) ÷ 3 which becomes eq.3a) 3×D = A + E + F
Hint #6
Substitute B – 2 for A (from eq.1a), 2 for E, and 2×B for F in eq.3a: 3×D = B – 2 + 2 + 2×B which becomes 3×D = 3×B Divide both sides by 3: 3×D ÷ 3 = 3×B ÷ 3 which makes D = B
Hint #7
Substitute B for D, 2×B for F, and (3×B – 6) for C (from eq.6a) in eq.5: B + 2×B = B × (3×B – 6) which becomes 3×B = 3×B² – 6×B which makes 9×B = 3×B² Since B ≠ 0 (from eq.4), divide both sides of the above equation by 3×B: 9×B ÷ 3×B = 3×B² ÷ 3×B which means 3 = B
Solution
Since B = 3, then: A = B – 2 = 3 – 2 = 1 (from eq.1a) C = 3×B – 6 = 3×3 – 6 = 9 – 6 = 3 (from eq.6a) D = B = 3 F = 2×B = 2×3 = 6 and ABCDEF = 133326