Puzzle for June 15, 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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Add D to both sides of eq.2: C - D + D = B + F + D which becomes eq.2a) C = B + F + D
Hint #2
In eq.4, replace C with B + F + D (from eq.2a): A + B + F + D = B + D + E Subtract both B and D from each side of the equation above: A + B + F + D - B - D = B + D + E - B - D which simplifies to eq.4a) A + F = E Subtract F from each side of eq.4a: A + F - F = E - F which becomes eq.4b) A = E - F
Hint #3
In eq.5, replace A with E - F (from eq.4b): B + E + F = E - F + D - F which becomes B + E + F = E - 2×F + D In the above equation, subtract E from both sides, and add 2×F to both sides: B + E + F - E + 2×F = E - 2×F + D - E + 2×F which becomes eq.5a) B + 3×F = D
Hint #4
eq.6 may be expressed as: 10×E + F = 10×A + B + C + D In the above equation, replace C with B + F + D (from eq.2a): 10×E + F = 10×A + B + B + D + F + D which is equivalent to 10×E + F = 10×A + 2×B + 2×D + F Subtract F from both sides: 10×E + F - F = 10×A + 2×B + 2×D + F - F which becomes eq.6a) 10×E = 10×A + 2×B + 2×D
Hint #5
In eq.6a, substitute (E - F) for A (from eq.4b): 10×E = 10×(E - F) + 2×B + 2×D which is the same as 10×E = 10×E - 10×F + 2×B + 2×D In the above equation, add 10×F to both sides, and subtract 10×E from both sides: 10×E + 10×F - 10×E = 10×E - 10×F + 2×B + 2×D + 10×F - 10×E which becomes 10×F = 2×B + 2×D Divide each side by 2: 10×F ÷ 2 = (2×B + 2×D) ÷ 2 which becomes eq.6b) 5×F = B + D
Hint #6
Substitute B + 3×F for D (from eq.5a) in eq.6b: 5×F = B + B + 3×F Subtract 3×F from each side: 5×F - 3×F = B + B + 3×F - 3×F which makes 2×F = 2×B Divide each side by 2: 2×F ÷ 2 = 2×B ÷ 2 which makes F = B
Hint #7
In eq.5a, replace F with B: B + 3×B = D which makes 4×B = D
Hint #8
Substitute B for F, and 4×B for D in eq.2a: C = B + B + 4×B which makes C = 6×B
Hint #9
Substitute B for F in eq.3: B - B = A - C which becomes 0 = A - C Add C to each side: 0 + C = A - C + C which means C = A which also means C = A = 6×B
Hint #10
Substitute 6×B for A, and B for F in eq.4a: 6×B + B = E which makes 7×B = E
Solution
Substitute 6×B for A and C, 4×B for D, 7×B for E, and B for F in eq.1: 6×B + B + 6×B + 4×B + 7×B + B = 25 which simplifies to 25×B = 25 Divide both sides by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = C = 6×B = 6 × 1 = 6 D = 4×B = 4 × 1 = 4 E = 7×B = 7 × 1 = 7 F = B = 1 and ABCDEF = 616471