Puzzle for March 6, 2019 ( )
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.
* BC and CD are 2-digit numbers (not B×C or C×D).
Scratchpad
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Hint #1
In eq.3, replace A with B - D + F (from eq.4): B - D + F + E = B + C + F Subtract both B and F from each side of the above equation: B - D + F + E - B - F = B + C + F - B - F which simplifies to -D + E = C which is equivalent to eq.3a) E - D = C In eq.2, replace E - D with C: B = C
Hint #2
Multiply both sides of eq.3a by (-1): (-1) × (E - D) = (-1) × C which becomes -E + D = -C which may be written as eq.3b) D - E = -C
Hint #3
In eq.5, substitute (-C) for D - E (from eq.3b): A - B - C + (-C) = B + C which becomes A - B - 2×C = B + C Substitute B for C in the above equation: A - B - 2×B = B + B which becomes A - 3×B = 2×B Add 3×B to both sides: A - 3×B + 3×B = 2×B + 3×B which makes A = 5×B
Hint #4
eq.6 may be re-stated as: 10×B + C - (10×C + D) = E which is the same as 10×B + C - 10×C - D = E which becomes 10×B - 9×C - D = E Substitute C for B in the above equation: 10×C - 9×C - D = E which becomes eq.6a) C - D = E
Hint #5
In eq.3b, substitute (C - D) for E (from eq.6a): D - (C - D) = -C which is equivalent to D - C + D = -C which becomes 2×D - C = -C Add C to both sides: 2×D - C + C = -C + C which becomes 2×D = 0 which means D = 0
Hint #6
Substitute 5×B for A, B for C, and 0 for D in eq.5: 5×B - B - B + 0 - E = B + B which simplifies to 3×B - E = 2×B Add (E - 2×B) to both sides: 3×B - E + (E - 2×B) = 2×B + (E - 2×B) which means B = E
Hint #7
Substitute 0 for D, and 5×B for A in eq.4: B - 0 + F = 5×B Subtract B from each side: B - 0 + F - B = 5×B - B which makes F = 4×B
Solution
Substitute 5×B for A, B for C and E, 0 for D, and 4×B for F in eq.1: 5×B + B + B + 0 + B + 4×B = 12 which becomes 12×B = 12 Divide both sides by 12: 12×B ÷ 12 = 12 ÷ 12 which means B = 1 making A = 5×B = 5 × 1 = 5 C = E = B = 1 F = 4×B = 4 × 1 = 4 and ABCDEF = 511014