Puzzle for February 16, 2022 ( )
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.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
In eq.3, replace C + E with B + D + F (from eq.4): B + D + F + F = A + B which becomes B + D + 2×F = A + B Subtract B from each side of the above equation: B + D + 2×F – B = A + B – B which becomes eq.3a) D + 2×F = A
Hint #2
In eq.2, replace A with D + 2×F (from eq.3a): D + E = D + 2×F – E In the above equation, subtract D from both sides, and add E to both sides: D + E – D + E = D + 2×F – E – D + E which simplifies to 2×E = 2×F Divide both sides by 2: 2×E ÷ 2 = 2×F ÷ 2 which makes E = F
Hint #3
In eq.5, substitute F for E, and (D + 2×F) for A (from eq.3a): F + F = C + D – (D + 2×F) – F which becomes 2×F = C + D – D – 2×F – F which becomes 2×F = C – 3×F Add 3×F to both sides of the above equation: 2×F + 3×F = C – 3×F + 3×F which makes 5×F = C
Hint #4
eq.6 may be written as: C + D + E = B + 10×E + F Substitute 5×F for C, and F for E in the above equation: 5×F + D + F = B + 10×F + F which becomes 6×F + D = B + 11×F Subtract 6×F from each side: 6×F + D – 6×F = B + 11×F – 6×F which becomes eq.6a) D = B + 5×F
Hint #5
Substitute B + 5×F for D (from eq.6a), 5×F for C, and F for E in eq.4: B + B + 5×F + F = 5×F + F which becomes 2×B + 6×F = 6×F Subtract 6×F from both sides of the above equation: 2×B + 6×F – 6×F = 6×F – 6×F which makes 2×B = 0 which means B = 0
Hint #6
Substitute 0 for B in eq.6a: D = 0 + 5×F which makes D = 5×F
Hint #7
Substitute 5×F for D in eq.3a: 5×F + 2×F = A which makes 7×F = A
Solution
Substitute 7×F for A, 0 for B, 5×F for C and D, and F for E in eq.1: 7×F + 0 + 5×F + 5×F + F + F = 19 which simplifies to 19×F = 19 Divide both sides of the above equation by 19: 19×F ÷ 19 = 19 ÷ 19 which means F = 1 making A = 7×F = 7 × 1 = 7 C = D = 5×F = 5 × 1 = 5 E = F = 1 and ABCDEF = 705511