Puzzle for December 29, 2023 ( )
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.
Scratchpad
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Hint #1
Add E to both sides of eq.5: D - C + E = B - E + F + E which becomes D - C + E = B + F which may be written as eq.5a) D + E - C = B + F
Hint #2
In eq.5a, replace D + E with A + C + F (from eq.4): A + C + F - C = B + F which becomes A + F = B + F Subtract F from each side of the equation above: A + F - F = B + F - F which makes A = B
Hint #3
In eq.2, replace B with A: A + E = A + D Subtract A from each side of the above equation: A + E - A = A + D - A which makes E = D
Hint #4
In eq.6, substitute D + E - C for B + F (from eq.5a): A + D + E - C = C + D + E In the equation above, subtract D and E from both sides, and add C to both sides: A + D + E - C - D - E + C = C + D + E - D - E + C which simplifies to A = 2×C and makes A = B = 2×C
Hint #5
Substitute 2×C for B in eq.3: E - F = F - 2×C Add F to both sides of the above equation: E - F + F = F - 2×C + F which makes E = 2×F - 2×C and also makes eq.3a) D = E = 2×F - 2×C
Hint #6
Substitute 2×C for A and B, and 2×F - 2×C for D and E (from eq.3a) in eq.1: 2×C + 2×C + 2×F - 2×C + C + 2×F - 2×C + F = 38 which simplifies to eq.1a) C + 5×F = 38
Hint #7
eq.1 may be written as: A + B + F + C + D + E = 38 Substitute A + B + F for C + D + E (from eq.6) into the above equation: A + B + F + A + B + F = 38 which may be written as eq.1b) 2×(A + B + F) = 38
Hint #8
In eq.1b, substitute 2×C for A and B, and C + 5×F for 38 (from eq.1a): 2×(2×C + 2×C + F) = C + 5×F which becomes 2×(4×C + F) = C + 5×F which becomes 8×C + 2×F = C + 5×F Subtract 2×F and C from each side of the equation above: 8×C + 2×F - 2×F - C = C + 5×F - 2×F - C which makes 7×C = 3×F Divide both sides by 3: 7×C ÷ 3 = 3×F ÷ 3 which makes 2⅓×C = F
Hint #9
Substitute (2⅓×C) for F in eq.3a: D = E = 2×(2⅓×C) - 2×C which becomes D = E = 4⅔×C - 2×C which makes D = E = 2⅔×C
Solution
Substitute (2⅓×C) for F in eq.1a: C + 5×(2⅓×C) = 38 which becomes C + 11⅔×C = 38 which makes 12⅔×C = 38 Divide both sides of the above equation by 12⅔: 12⅔×C ÷ 12⅔ = 38 ÷ 12⅔ which means C = 3 making A = B = 2×C = 2 × 3 = 6 D = E = 2⅔×C = 2⅔ × 3 = 8 F = 2⅓×C = 2⅓ × 3 = 7 and ABCDEF = 663887