Puzzle for February 15, 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
Subtract A from each side of eq.6: D – A – A = A + C + E – A which becomes eq.6a) D – 2×A = C + E Subtract B from each side of eq.3: D – B = B + C + E – B which becomes eq.3a) D – B = C + E
Hint #2
In eq.3a, replace C + E with D – 2×A (from eq.6a): D – B = D – 2×A Subtract D from each side of the equation above: D – B – D = D – 2×A – D which becomes –B = –2×A Multiply both sides by (–1): –B × (–1) = –2×A × (–1) which makes B = 2×A
Hint #3
In eq.2, replace B with 2×A: F = A + 2×A which makes F = 3×A
Hint #4
In eq.5, substitute 3×A for F, and 2×A for B: E + 3×A = A + 2×A + C which becomes E + 3×A = 3×A + C Subtract 3×A from both sides of the above equation: E + 3×A – 3×A = 3×A + C – 3×A which makes E = C
Hint #5
In eq.4, substitute 2×A for B, and 3×A for F: 2×A + 3×A = D – 3×A which becomes 5×A = D – 3×A Add 3×A to both sides of the equation above: 5×A + 3×A = D – 3×A + 3×A which makes 8×A = D
Hint #6
Substitute 8×A for D, 2×A for B, and C for E in eq.3: 8×A = 2×A + C + C which becomes 8×A = 2×A + 2×C Subtract 2×A from both sides of the above equation: 8×A – 2×A = 2×A + 2×C – 2×A which makes 6×A = 2×C Divide both sides by 2: 6×A ÷ 2 = 2×C ÷ 2 which makes 3×A = C and also makes E = C = 3×A
Solution
Substitute 2×A for B, 3×A for C and E and F, and 8×A for D in eq.1: A + 2×A + 3×A + 8×A + 3×A + 3×A = 20 which simplifies to 20×A = 20 Divide both sides of the above equation by 20: 20×A ÷ 20 = 20 ÷ 20 which means A = 1 making B = 2×A = 2 × 1 = 2 C = E = F = 3×A = 3 × 1 = 3 D = 8×A = 8 × 1 = 8 and ABCDEF = 123833