Puzzle for December 2, 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.
* AB and DE are 2-digit numbers (not A×B or D×E). BCD is a 3-digit number (not B×C×D).
Scratchpad
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Hint #1
eq.6 may be written as: B = (A + C + E) ÷ 3 Multiply both sides of the above equation by 3: B × 3 = ((A + C + E) ÷ 3) × 3 which becomes eq.6a) 3×B = A + C + E
Hint #2
In eq.6a, replace A with B + E (from eq.4): 3×B = B + E + C + E which becomes 3×B = B + C + 2×E Subtract B from each side of the above equation: 3×B – B = B + C + 2×E – B which becomes eq.6b) 2×B = C + 2×E
Hint #3
In eq.6b, substitute (E – C) for B (from eq.2): 2×(E – C) = C + 2×E which becomes 2×E – 2×C = C + 2×E In the above equation, subtract 2×E from both sides, and add 2×C to both sides: 2×E – 2×C – 2×E + 2×C = C + 2×E – 2×E + 2×C which simplifies to 0 = 3×C which means 0 = C
Hint #4
Substitute 0 for C in eq.2: E – 0 = B which makes E = B
Hint #5
Substitute B for E in eq.4: A = B + B which makes A = 2×B
Hint #6
eq.5 may be written as: 100×B + 10×C + D + E = 10×A + B + 10×D + E + F Subtract B, D, and E from each side of the above equation: 100×B + 10×C + D + E – B – D – E = 10×A + B + 10×D + E + F – B – D – E which becomes eq.5a) 99×B + 10×C = 10×A + 9×D + F
Hint #7
In eq.5a, substitute 0 for C, (2×B) for A, and (B + F) for D (from eq.3): 99×B + 10×0 = 10×(2×B) + 9×(B + F) + F which becomes 99×B = 20×B + 9×B + 9×F + F which becomes 99×B = 29×B + 10×F Subtract 29×B from each side of the equation above: 99×B – 29×B = 29×B + 10×F – 29×B which becomes 70×B = 10×F Divide both sides by 10: 70×B ÷ 10 = 10×F ÷ 10 which becomes 7×B = F
Hint #8
Substitute 7×B for F into eq.3: D = B + 7×B which makes D = 8×B
Solution
Substitute 2×B for A, 0 for C, 8×B for D, B for E, 7×B for F in eq.1: 2×B + B + 0 + 8×B + B + 7×B = 19 which simplifies to 19×B = 19 Divide both sides of the above equation by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = 2×B = 2 × 1 = 2 D = 8×B = 8 × 1 = 8 E = B = 1 F = 7×B = 7 × 1 = 7 and ABCDEF = 210817