Puzzle for March 9, 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 A + B with F (from eq.6): F + F = C which makes 2×F = C
Hint #2
In eq.2, replace F with A + B (from eq.6): B + A + B = A + E which becomes 2×B + A = A + E Subtract A from both sides of the equation above: 2×B + A – A = A + E – A which makes eq.2a) 2×B = E
Hint #3
In eq.3, substitute A + B + F for C (from eq.4), and 2×B for E (from eq.2a): A + B + F + 2×B = A + D + F which becomes A + F + 3×B = A + D + F Subtract A and F from both sides of the above equation: A + F + 3×B – A – F = A + D + F – A – F which simplifies to eq.3a) 3×B = D
Hint #4
Substitute 3×B for D (from eq.3a), 2×B for E (from eq.2a), and 2×F for C in eq.5: 3×B + 2×B = B + 2×F + F which becomes 5×B = B + 3×F Subtract B from each side of the equation above: 5×B – B = B + 3×F – B which makes 4×B = 3×F Divide both sides by 4: 4×B ÷ 4 = 3×F ÷ 4 which makes B = ¾×F
Hint #5
Substitute (¾×F) for B in eq.3a: 3×(¾×F) = D which makes 2¼×F = D
Hint #6
Substitute (¾×F) for B in eq.2a: 2×(¾×F) = E which makes 1½×F = E
Hint #7
Substitute ¾×F for B in eq.6: F = A + ¾×F Subtract ¾×F from each side of the equation above: F – ¾×F = A + ¾×F – ¾×F which makes ¼×F = A
Solution
Substitute ¼×F for A, ¾×F for B, 2×F for C, 2¼×F for D, and 1½×F for E in eq.1: ¼×F + ¾×F + 2×F + 2¼×F + 1½×F + F = 31 which simplifies to 7¾×F = 31 Divide both sides of the above equation by 7¾: 7¾×F ÷ 7¾ = 31 ÷ 7¾ which means F = 4 making A = ¼×F = ¼ × 4 = 1 B = ¾×F = ¾ × 4 = 3 C = 2×F = 2 × 4 = 8 D = 2¼×F = 2¼ × 4 = 9 E = 1½×F = 1½ × 4 = 6 and ABCDEF = 138964