Puzzle for June 12, 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.
* CD and EF are 2-digit numbers (not C×D or E×F).
Scratchpad
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Hint #1
Add A to both sides of eq.2: B + A = F - A + A which becomes B + A = F which is the same as eq.2a) A + B = F
Hint #2
In eq.5, replace A + B with F (from eq.2a): C + F = F + E Subtract F from both sides of the equation above: C + F - F = F + E - F which becomes C = E
Hint #3
eq.6 may be written as: 10×E + F = A - B + 10×C + D In the equation above, replace E with C: 10×C + F = A - B + 10×C + D Subtract 10×C from each side: 10×C + F - 10×C = A - B + 10×C + D - 10×C which becomes eq.6a) F = A - B + D
Hint #4
In eq.6a, substitute A + B for F (from eq.2a): A + B = A - B + D In the above equation, add B to both sides, and subtract A from both sides: A + B + B - A = A - B + D + B - A which simplifies to 2×B = D
Hint #5
In eq.4, add D to each side, and subtract A from each side: A - D + F + D - A = D + D - A which becomes F = 2×D - A Substitute (2×D - A) for F in eq.2: B = (2×D - A) - A which becomes eq.4a) B = 2×D - 2×A
Hint #6
Substitute (2×B) for D in eq.4a: B = 2×(2×B) - 2×A which becomes B = 4×B - 2×A Add 2×A to each side, and subtract B from each side: B + 2×A - B = 4×B - 2×A + 2×A - B which simplifies to 2×A = 3×B Divide each side by 2: 2×A ÷ 2 = 3×B ÷ 2 which makes A = 1½×B
Hint #7
Substitute 1½×B for A in eq.2a: 1½×B + B = F which means 2½×B = F
Hint #8
Substitute 2×B for D, and 2½×B for F in eq.3: 2×B = E - 2½×B Add 2½×B to each side of the above equation: 2×B + 2½×B = E - 2½×B + 2½×B which makes 4½×B = E which also makes C = E = 4½×B
Solution
Substitute 1½×B for A, 4½×B for C and E, 2×B for D, and 2½×B for F in eq.1: 1½×B + B + 4½×B + 2×B + 4½×B + 2½×B = 32 which simplifies to 16×B = 32 Divide both sides of the above equation by 16: 16×B ÷ 16 = 32 ÷ 16 which means B = 2 making A = 1½×B = 1½ × 2 = 3 C = E = 4½×B = 4½ × 2 = 9 D = 2×B = 2 × 2 = 4 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 329495