Puzzle for April 30, 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
In eq.5, replace B + C with A + F (from eq.3): A + E = A + F + F which becomes A + E = A + 2×F Subtract A from both sides of the above equation: A + E – A = A + 2×F – A which makes E = 2×F
Hint #2
In eq.6, replace B + D with E + F (from eq.4): C + E = E + F + F which becomes C + E = E + 2×F Subtract E from both sides of the equation above: C + E – E = E + 2×F – E which makes C = 2×F and also makes eq.6a) C = E = 2×F
Hint #3
In eq.5, substitute 2×F for both E and C: A + 2×F = B + 2×F + F Subtract 2×F from each side of the equation above: A + 2×F – 2×F = B + 2×F + F – 2×F which becomes eq.5a) A = B + F
Hint #4
Substitute 2×F for E, and B + F for A (from eq.5a) in eq.2: D + 2×F = B + F + B which becomes D + 2×F = 2×B + F Subtract 2×F from both sides of the above equation: D + 2×F – 2×F = 2×B + F – 2×F which becomes eq.2a) D = 2×B – F
Hint #5
Substitute 2×F for E, and 2×B – F for D (from eq.2a) in eq.4: 2×F + F = B + 2×B – F which becomes 3×F = 3×B – F Add F to both sides of the equation above: 3×F + F = 3×B – F + F which makes 4×F = 3×B Divide both sides by 4: 4×F ÷ 4 = 3×B ÷ 4 which makes F = ¾×B
Hint #6
Substitute ¾×B for F in eq.5a: A = B + ¾×B which makes A = 1¾×B
Hint #7
Substitute (¾×B) for F in eq.6a: C = E = 2×(¾×B) which makes C = E = 1½×B
Hint #8
Substitute ¾×B for F in eq.2a: D = 2×B – ¾×B which makes D = 1¼×B
Solution
Substitute 1¾×B for A, 1½×B for C and E, 1¼×B for D, and ¾×B for F in eq.1: 1¾×B + B + 1½×B + 1¼×B + 1½×B + ¾×B = 31 which simplifies to 7¾×B = 31 Divide both sides of the above equation by 7¾: 7¾×B ÷ 7¾ = 31 ÷ 7¾ which means B = 4 making A = 1¾×B = 1¾ × 4 = 7 C = E = 1½×B = 1½ × 4 = 6 D = 1¼×B = 1¼ × 4 = 5 F = ¾×B = ¾ × 4 = 3 and ABCDEF = 746563