Puzzle for December 9, 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
Subtract D and F from both sides of eq.4: A + D – D – F = B + F – D – F which becomes eq.4a) A – F = B – D Subtract F and E from both sides of eq.5: D + F – F – E = A + E – F – E which becomes eq.5a) D – E = A – F
Hint #2
In eq.4a, replace A – F with D – E (from eq.5a): D – E = B – D Add D to both sides of the above equation: D – E + D = B – D + D which becomes eq.4b) 2×D – E = B
Hint #3
In eq.4b, replace B with D + E (from eq.2): 2×D – E = D + E In the above equation, add E to both sides, and subtract D from both sides: 2×D – E + E – D = D + E + E – D which makes D = 2×E
Hint #4
In eq.4b, substitute (2×E) for D: 2×(2×E) – E = B which becomes 4×E – E = B which makes 3×E = B
Hint #5
Substitute 2×E for D, and 3×E for B in eq.3: C – 2×E = 3×E – C + 2×E which makes 2×C = 7×E Divide both sides of the above equation by 2: 2×C ÷ 2 = 7×E ÷ 2 which makes C = 3½×E
Hint #6
Substitute 3½×E for C, 3×E for B, and 2×E for D in eq.6: 3½×E – A = A + 3×E – 3½×E + 2×E which becomes 3½×E – A = A + 1½×E In the above equation, add A to both sides, and subtract 1½×E from both sides: 3½×E – A + A – 1½×E = A + 1½×E + A – 1½×E which becomes 2×E = 2×A Divide both sides by 2: 2×E ÷ 2 = 2×A ÷ 2 which makes E = A
Hint #7
Substitute 2×E for D, and E for A in eq.5: 2×E + F = E + E which makes 2×E + F = 2×E Subtract 2×E from both sides of the equation above: 2×E + F – 2×E = 2×E – 2×E which becomes F = 0
Solution
Substitute E for A, 3×E for B, 3½×E for C, 2×E for D, and 0 for F in eq.1: E + 3×E + 3½×E + 2×E + E + 0 = 21 which simplifies to 10½×E = 21 Divide both sides of the above equation by 10½: 10½×E ÷ 10½ = 21 ÷ 10½ which means E = 2 making A = E = 2 B = 3×E = 3 × 2 = 6 C = 3½×E = 3½ × 2 = 7 D = 2×E = 2 × 2 = 4 and ABCDEF = 267420