Puzzle for April 28, 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
Add F to both sides of eq.5: B + D + F = A + C + E + F In the above equation, replace D + F with A + E (from eq.3): B + A + E = A + C + E + F Subtract A and E from each side: B + A + E – A – E = A + C + E + F – A – E which becomes eq.5a) B = C + F
Hint #2
In eq.2, replace B with C + F (from eq.5a): C + F = E + F Subtract F from each side of the above equation: C + F – F = E + F – F which becomes C = E
Hint #3
Substitute (E + F) for B (from eq.2), and E for C in eq.6: E – F = A – (E + F) + E which becomes E – F = A – E – F + E which becomes E – F = A – F Add F to both sides of the equation above: E – F + F = A – F + F which makes E = A
Hint #4
Substitute (E + F) for B (from eq.2) in eq.4: F – D = D – (E + F) which becomes F – D = D – E – F Add D, E, and F to both sides of the above equation: F – D + D + E + F = D – E – F + D + E + F which becomes E + 2×F = 2×D Divide both sides by 2: (E + 2×F) ÷ 2 = 2×D ÷ 2 which becomes eq.4a) ½×E + F = D
Hint #5
Substitute ½×E + F for D (from eq.4a), and E for A in eq.3: ½×E + F + F = E + E which becomes ½×E + 2×F = 2×E Subtract ½×E from each side of the above equation: ½×E + 2×F – ½×E = 2×E – ½×E which makes 2×F = 1½×E Divide both sides by 2: 2×F ÷ 2 = 1½×E ÷ 2 which makes F = ¾×E
Hint #6
Substitute ¾×E for F in eq.4a: ½×E + ¾×E = D which makes 1¼×E = D
Hint #7
Substitute ¾×E for F in eq.2: B = E + ¾×E which makes B = 1¾×E
Solution
Substitute E for A and C, 1¾×E for B, 1¼×E for D, and ¾×E for F in eq.1: E + 1¾×E + E + 1¼×E + E + ¾×E = 27 which simplifies to 6¾×E = 27 Divide both sides of the above equation by 6¾: 6¾×E ÷ 6¾ = 27 ÷ 6¾ which means E = 4 making A = C = E = 4 B = 1¾×E = 1¾ × 4 = 7 D = 1¼×E = 1¼ × 4 = 5 F = ¾×E = ¾ × 4 = 3 and ABCDEF = 474543