Puzzle for April 30, 2021 ( )
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.6, replace D + E with A (from eq.2): A + C – E = B + A Subtract A from both sides of the above equation: A + C – E – A = B + A – A which becomes eq.6a) C – E = B
Hint #2
In eq.3, replace B with C – E (from eq.6a): C + E = A + C – E In the equation above, subtract C from both sides, and add E to both sides: C + E – C + E = A + C – E – C + E which makes 2×E = A
Hint #3
In eq.2, substitute 2×E for A: D + E = 2×E Subtract E from each side of the above equation: D + E – E = 2×E – E which makes D = E
Hint #4
Substitute E for D, and 2×E for A in eq.4: C + E – F = 2×E + E Subtract E from both sides of the equation above: C + E – F – E = 2×E + E – E which becomes eq.4a) C – F = 2×E
Hint #5
Substitute C – E for B (from eq.6a), 2×E for A, and E for D in eq.5: C – E + F – 2×E = 2×E + E + E which becomes C – 3×E + F = 4×E Add 3×E to both sides of the above equation: C – 3×E + F + 3×E = 4×E + 3×E which becomes eq.5a) C + F = 7×E
Hint #6
Add the left and right sides of eq.4a to the left and right sides of eq.5a, respectively: C + F + C – F = 7×E + 2×E which becomes 2×C = 9×E Divide both sides of the above equation by 2: 2×C ÷ 2 = 9×E ÷ 2 which makes C = 4½×E
Hint #7
Substitute 4½×E for C in eq.6a: 4½×E – E = B which makes 3½×E = B
Hint #8
Substitute 3½×E for B in eq.5a: 3½×E + F = 6×E Subtract 3½×E from each side of the equation above: 3½×E + F – 3½×E = 6×E – 3½×E which makes F = 2½×E
Solution
Substitute 2×E for A, 3½×E for B, 4½×E for C, E for D, and 2½×E for F in eq.1: 2×E + 3½×E + 4½×E + E + E + 2½×E = 29 which simplifies to 14½×E = 29 Divide both sides of the above equation by 14½: 14½×E ÷ 14½ = 29 ÷ 14½ which means E = 2 making A = 2×E = 2 × 2 = 4 B = 3½×E = 3½ × 2 = 7 C = 4½×E = 4½ × 2 = 9 D = E = 2 F = 2½×E = 2½ × 2 = 5 and ABCDEF = 479225