Puzzle for February 22, 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.3, replace F with (C – B) from eq.4: A = D – (C – B) which becomes A = D – C + B Subtract B from both sides of the above equation: A – B = D – C + B – B which becomes eq.4a) A – B = D – C
Hint #2
In eq.5, replace A – B with D – C (from eq.4a): D – E = D – C Subtract D from each side of the equation above: D – E – D = D – C – D which makes –E = –C which means E = C
Hint #3
Multiply both sides of eq.5 by (–1): (D – E) × (–1) = (A – B) × (–1) which means E – D = B – A In eq.6, substitute B – A for E – D: F – (B – A) = B – A Add (B – A) to both sides of the above equation: F – (B – A) + (B – A) = B – A + (B – A) which becomes eq.6a) F = 2×B – 2×A
Hint #4
Add F to each side of eq.2: B – F + F = F – A + F which becomes eq.2a) B = 2×F – A
Hint #5
In eq.6a, substitute (2×F – A) for B (from eq.2a): F = 2×(2×F – A) – 2×A which becomes F = 4×F – 2×A – 2×A which becomes F = 4×F – 4×A In the equation above, add 4×A to each side, and subtract F from both sides: F + 4×A – F = 4×F – 4×A + 4×A – F which makes 4×A = 3×F Divide both sides by 4: 4×A ÷ 4 = 3×F ÷ 4 which means A = ¾×F
Hint #6
Substitute ¾×F for A in eq.3: ¾×F = D – F Add F to both sides of the above equation: ¾×F + F = D – F + F which makes 1¾×F = D
Hint #7
Substitute ¾×F for A in eq.2a: B = 2×F – ¾×F which makes B = 1¼×F
Hint #8
Substitute 1¼×F for B in eq.4: C – 1¼×F = F Add 1¼×F to both sides of the above equation: C – 1¼×F + 1¼×F = F + 1¼×F which means C = 2¼×F and also means E = C = 2¼×F
Solution
Substitute ¾×F for A, 1¼×F for B, 2¼×F for C and E, and 1¾×F for D in eq.1: ¾×F + 1¼×F + 2¼×F + 1¾×F + 2¼×F + F = 37 which simplifies to 9¼×F = 37 Divide both sides of the equation above by 9¼: 9¼×F ÷ 9¼ = 37 ÷ 9¼ which means F = 4 making A = ¾×F = ¾ × 4 = 3 B = 1¼×F = 1¼ × 4 = 5 C = E = 2¼×F = 2¼ × 4 = 9 D = 1¾×F = 1¾ × 4 = 7 and ABCDEF = 359794