Puzzle for August 21, 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
eq.6 may be written as: B = (A + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × B = 4 × ((A + C + D + F) ÷ 4) which becomes eq.6a) 4×B = A + C + D + F
Hint #2
In eq.6a, substitute (C + D) for B (from eq.2): 4×(C + D) = A + C + D + F which may be written as 4×C + 4×D = A + C + D + F Subtract C and D from both sides of the above equation: 4×C + 4×D – C – D = A + C + D + F – C – D which becomes eq.6b) 3×C + 3×D = A + F
Hint #3
eq.3 may be written as: B + C = A + F – C – D In the above equation, replace B with C + D (from eq.2), and replace A + F with 3×C + 3×D (from eq.6b): C + D + C = 3×C + 3×D – C – D which becomes 2×C + D = 2×C + 2×D Subtract 2×C and D from both sides: 2×C + D – 2×C – D = 2×C + 2×D – 2×C – D which simplifies to 0 = D
Hint #4
In eq.2, replace D with 0: C + 0 = B which makes C = B
Hint #5
eq.5 may be written as: E = (A + B) ÷ 2 Multiply both sides of the above equation by 2: 2 × E = 2 × ((A + B) ÷ 2) which becomes eq.5a) 2×E = A + B
Hint #6
eq.4 may be written as: A = (E + F) ÷ 2 Multiply both sides of the above equation by 4: 4 × A = 4 × ((E + F) ÷ 2) which becomes 4×A = 2×E + 2×F Subtract 2×F from each side: 4×A – 2×F = 2×E + 2×F – 2×F which becomes eq.4a) 4×A – 2×F = 2×E
Hint #7
In eq.5a, substitute 4×A – 2×F for 2×E (from eq.4a), and C for B: 4×A – 2×F = A + C Subtract A from both sides of the equation above: 4×A – 2×F – A = A + C – A which becomes eq.5b) 3×A – 2×F = C
Hint #8
In eq.6b, substitute (3×A – 2×F) for C (from eq.5b), and 0 for D: 3×(3×A – 2×F) + 3×0 = A + F which becomes 9×A – 6×F = A + F In the above equation, add 6×F to both sides, and subtract A from both sides: 9×A – 6×F + 6×F – A = A + F + 6×F – A which makes 8×A = 7×F Divide both sides by 8: 8×A ÷ 8 = 7×F ÷ 8 which makes A = ⅞×F
Hint #9
Substitute (⅞×F) for A in eq.5b: 3×(⅞×F) – 2×F = C which becomes 2⅝×F – 2×F = C which makes ⅝×F = C and also makes B = C = ⅝×F
Hint #10
Substitute ⅞×F for A, and ⅝×F for B in eq.5a: 2×E = ⅞×F + ⅝×F which makes 2×E = 1½×F Divide both sides of the above equation by 2: 2×E ÷ 2 = 1½×F ÷ 2 which makes E = ¾×F
Solution
Substitute ⅞×F for A, ⅝×F for B and C, 0 for D, and ¾×F for E in eq.1: ⅞×F + ⅝×F + ⅝×F + 0 + ¾×F + F = 31 which simplifies to 3⅞×F = 31 Divide both sides of the equation above by 3⅞: 3⅞×F ÷ 3⅞ = 31 ÷ 3⅞ which means F = 8 making A = ⅞×F = ⅞ × 8 = 7 B = C = ⅝×F = ⅝ × 8 = 5 E = ¾×F = ¾ × 8 = 6 and ABCDEF = 755068