Puzzle for February 28, 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.4, replace B with D + E (from eq.2): C + D = A + D + E Subtract D from both sides of the above equation: C + D – D = A + D + E – D which becomes eq.4a) C = A + E
Hint #2
In eq.3, replace A + E with C (from eq.4a): C = C + F Subtract C from both sides of the above equation: C – C = C + F – C which makes 0 = F
Hint #3
eq.6 may be written as: B – A = C – B – B + C which becomes B – A = 2×C – 2×B Add A and 2×B to both sides of the above equation: B – A + A + 2×B = 2×C – 2×B + A + 2×B which becomes 3×B = 2×C + A Subtract 2×C from both sides: 3×B – 2×C = 2×C + A – 2×C which becomes eq.6a) 3×B – 2×C = A
Hint #4
In eq.4, substitute 3×B – 2×C for A (from eq.6a): C + D = 3×B – 2×C + B which becomes C + D = 4×B – 2×C Subtract C from both sides of the above equation: C + D – C = 4×B – 2×C – C which becomes eq.4b) D = 4×B – 3×C
Hint #5
Substitute (3×B – 2×C) for A (from eq.6a), 4×B – 3×C for D (from eq.4b), and 0 for F in eq.5: B + C – (3×B – 2×C) = (3×B – 2×C) + 4×B – 3×C + 0 which becomes B + C – 3×B + 2×C = 7×B – 5×C which becomes 3×C – 2×B = 7×B – 5×C Add 2×B and 5×C to both sides of the equation above: 3×C – 2×B + 2×B + 5×C = 7×B – 5×C + 2×B + 5×C which simplifies to 8×C = 9×B Divide both sides by 8: 8×C ÷ 8 = 9×B ÷ 8 which makes C = 1⅛×B
Hint #6
Substitute (1⅛×B) for C in eq.6a: 3×B – 2×(1⅛×B) = A which becomes 3×B – 2¼×B = A which makes ¾×B = A
Hint #7
Substitute (1⅛×B) for C in eq.4b: D = 4×B – 3×(1⅛×B) which becomes D = 4×B – 3⅜×B which makes D = ⅝×B
Hint #8
Substitute ⅝×B for D in eq.2: ⅝×B + E = B Subtract ⅝×B from both sides of the equation above: ⅝×B + E – ⅝×B = B – ⅝×B which makes E = ⅜×B
Solution
Substitute ¾×B for A, 1⅛×B for C, ⅝×B for D, ⅜×B for E, and 0 for F in eq.1: ¾×B + B + 1⅛×B + ⅝×B + ⅜×B + 0 = 31 which becomes 3⅞×B = 31 Divide both sides of the equation above by 3⅞: 3⅞×B ÷ 3⅞ = 31 ÷ 3⅞ which means B = 8 making A = ¾×B = ¾ × 8 = 6 C = 1⅛×B = 1⅛ × 8 = 9 D = ⅝×B = ⅝ × 8 = 5 E = ⅜×B = ⅜ × 8 = 3 and ABCDEF = 689530