Puzzle for February 18, 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.4, replace D with F – A (from eq.2): B – C = F – A – F + A which becomes B – C = 0 Add C to each side of the above equation: B – C + C = 0 + C which makes B = C
Hint #2
In eq.6, replace B with C: E + F – C = C + C which becomes E + F – C = 2×C In the above equation, add C to both sides, and subtract F from both sides: E + F – C + C – F = 2×C + C – F which becomes eq.6a) E = 3×C – F
Hint #3
In eq.3, substitute (3×C – F) for E (from eq.6a): D – C = F – (3×C – F) which is equivalent to D – C = F – 3×C + F which becomes D – C = 2×F – 3×C Add C to both sides of the above equation: D – C + C = 2×F – 3×C + C which becomes eq.3a) D = 2×F – 2×C
Hint #4
Substitute (2×F – 2×C) for D (from eq.3a), (3×C – F) for E (from eq.6a), and C for B in eq.5: (2×F – 2×C) – (3×C – F) = C – (2×F – 2×C) which becomes 2×F – 2×C – 3×C + F = C – 2×F + 2×C which becomes 3×F – 5×C = 3×C – 2×F Add 5×C and 2×F to both sides of the above equation: 3×F – 5×C + 5×C + 2×F = 3×C – 2×F + 5×C + 2×F which makes 5×F = 8×C Divide both sides by 8: 5×F ÷ 8 = 8×C ÷ 8 which makes ⅝×F = C and also makes B = C = ⅝×F
Hint #5
Substitute (⅝×F) for C in eq.3a: D = 2×F – 2×(⅝×F) which becomes D = 2×F – 1¼×F which makes D = ¾×F
Hint #6
Substitute (⅝×F) for C in eq.6a: E = 3×(⅝×F) – F which becomes E = 1⅞×F – F which makes E = ⅞×F
Hint #7
Substitute ¾×F for D in eq.2: F – A = ¾×F In the above equation, add A to both sides, and subtract ¾×F from both sides: F – A + A – ¾×F = ¾×F + A – ¾×F which makes ¼×F = A
Solution
Substitute ¼×F for A, ⅝×F for B and C, ¾×F for D, and ⅞×F for E in eq.1: ¼×F + ⅝×F + ⅝×F + ¾×F + ⅞×F + F = 33 which simplifies to 4⅛×F = 33 Divide both sides of the above equation by 4⅛: 4⅛×F ÷ 4⅛ = 33 ÷ 4⅛ which means F = 8 making A = ¼×F = ¼ × 8 = 2 B = C = ⅝×F = ⅝ × 8 = 5 D = ¾×F = ¾ × 8 = 6 E = ⅞×F = ⅞ × 8 = 7 and ABCDEF = 255678