Puzzle for May 20, 2020 ( )
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.3, replace B + C with A – B + E (from eq.6): A – C – E = A – B + E Subtract A from both sides of the above equation: A – C – E – A = A – B + E – A which becomes –C – E = –B + E Add B, C, and E to both sides: –C – E + B + C + E = –B + E + B + C + E which becomes eq.3a) B = C + 2×E
Hint #2
Add E to both sides of eq.5: A – E + E = B + E + E which becomes A = B + 2×E In the above equation, replace B with C + 2×E (from eq.3a): A = C + 2×E + 2×E which becomes eq.5a) A = C + 4×E
Hint #3
In eq.6, substitute C + 2×E for B (from eq.3a) and C + 4×E for A (from eq.5a): C + 2×E + C = C + 4×E – (C + 2×E) + E which becomes 2×C + 2×E = C + 5×E – C – 2×E which becomes 2×C + 2×E = 3×E Subtract 2×E from both sides of the above equation: 2×C + 2×E – 2×E = 3×E – 2×E which makes 2×C = E
Hint #4
Substitute (2×C) for E in eq.5a: A = C + 4×(2×C) which is equivalent to A = C + 8×C which makes A = 9×C
Hint #5
Substitute (2×C) for E in eq.3a: B = C + 2×(2×C) which is equivalent to B = C + 4×C which makes B = 5×C
Hint #6
Substitute 5×C for B in eq.4: F = 5×C – C which makes F = 4×C
Hint #7
Substitute 2×C for E, 4×C for F, and 5×C for B in eq.2: 2×C + 4×C = 5×C + C + D which becomes 6×C = 6×C + D Subtract 6×C from both sides of the equation above: 6×C – 6×C = 6×C + D – 6×C which means eq.2) 0 = D
Solution
Substitute 9×C for A, 5×C for B, 0 for D, 2×C for E, and 4×C for F in eq.1: 9×C + 5×C + C + 0 + 2×C + 4×C = 21 which simplifies to 21×C = 21 Divide both sides of the equation above by 21: 21×C ÷ 21 = 21 ÷ 21 which means C = 1 making A = 9×C = 9 × 1 = 9 B = 5×C = 5 × 1 = 5 E = 2×C = 2 × 1 = 2 F = 4×C = 4 × 1 = 4 and ABCDEF = 951024