Puzzle for August 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.4, replace A + B with D + E (from eq.3): C + D – E = D + E + E + F which becomes C + D – E = D + 2×E + F In the above equation, subtract D from both sides, and add E to both sides: C + D – E – D + E = D + 2×E + F – D + E which becomes eq.4a) C = 3×E + F
Hint #2
In eq.2, replace C with 3×E + F (from eq.4a): B + F = 3×E + F + E which becomes B + F = 4×E + F Subtract F from each side of the equation above: B + F – F = 4×E + F – F which makes B = 4×E
Hint #3
Add D to both sides of eq.6: B + C – D + D = A + E + F + D which becomes B + C = A + E + F + D which may be written as B + C = A + D + E + F In the equation above, substitute C + E + F for A + D (from eq.5): B + C = C + E + F + E + F which becomes B + C = C + 2×E + 2×F Subtract C from both sides: B + C – C = C + 2×E + 2×F – C which becomes eq.6a) B = 2×E + 2×F
Hint #4
Substitute 4×E for B in eq.6a: 4×E = 2×E + 2×F Subtract 2×E from both sides of the equation above: 4×E – 2×E = 2×E + 2×F – 2×E which makes 2×E = 2×F Divide both sides by 2: 2×E ÷ 2 = 2×F ÷ 2 which makes E = F
Hint #5
Substitute E for F in eq.4a: C = 3×E + E which makes C = 4×E
Hint #6
Subtract B from both sides of eq.3: D + E – B = A + B – B which becomes D + E – B = A Substitute D + E – B for A in eq.5: D + E – B + D = C + E + F which becomes 2×D + E – B = C + E + F Subtract E from both sides of the above equation: 2×D + E – B – E = C + E + F – E which becomes eq.5a) 2×D – B = C + F
Hint #7
Substitute 4×E for B and C, and E for F in eq.5a: 2×D – 4×E = 4×E + E which becomes 2×D – 4×E = 5×E Add 4×E to each side of the equation above: 2×D – 4×E + 4×E = 5×E + 4×E which makes 2×D = 9×E Divide both sides by 2: 2×D ÷ 2 = 9×E ÷ 2 which makes D = 4½×E
Hint #8
Substitute 4½×E for D, and 4×E for B in eq.3: 4½×E + E = A + 4×E which becomes 5½×E = A + 4×E Subtract 4×E from both sides of the above equation: 5½×E – 4×E = A + 4×E – 4×E which makes 1½×E = A
Solution
Substitute 1½×E for A, 4×E for B and C, 4½×E for D, and E for F in eq.1: 1½×E + 4×E + 4×E + 4½×E + E + E = 32 which simplifies to 16×E = 32 Divide both sides of the above equation by 16: 16×E ÷ 16 = 32 ÷ 16 which means E = 2 making A = 1½×E = 1½ × 2 = 3 B = C = 4×E = 4 × 2 = 8 D = 4½×E = 4½ × 2 = 9 F = E = 2 and ABCDEF = 388922