Puzzle for April 21, 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.
* CD is a 2-digit number (not C×D).
Scratchpad
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Hint #1
Add A and D to both sides of eq.4: F - A + A + D = C - D + A + D which becomes F + D = C + A which may be written as eq.4a) D + F = C + A
Hint #2
In eq.4a, replace F with C + D (from eq.2): D + C + D = C + A which becomes 2×D + C = C + A Subtract C from both sides of the above equation: 2×D + C - C = C + A - C which makes eq.4b) 2×D = A
Hint #3
eq.6 may be written as: 10×C + D = A + C + F Subtract C from each side of the equation above: 10×C + D - C = A + C + F - C which becomes eq.6a) 9×C + D = A + F
Hint #4
In eq.6a, replace A with 2×D: 9×C + D = 2×D + F which becomes Subtract D from each side of the above equation: 9×C + D - D = 2×D + F - D which becomes eq.6b) 9×C = D + F
Hint #5
In eq.4a, substitute 9×C for D + F (from eq.6b): 9×C = C + A Subtract C from each side of the above equation: 9×C - C = C + A - C which makes 8×C = A
Hint #6
Substitute 8×C for A in eq.4b: 2×D = 8×C Divide both sides of the above equation by 2: 2×D ÷ 2 = 8×C ÷ 2 which makes D = 4×C
Hint #7
Substitute 4×C for D in eq.2: F = C + 4×C which makes F = 5×C
Hint #8
Substitute 5×C for F, and 8×C for A in eq.3: E + 5×C = 8×C + B Subtract 5×C from both sides of the equation above: E + 5×C - 5×C = 8×C + B - 5×C which becomes eq.3a) E = 3×C + B
Hint #9
Substitute 5×C for F, and 8×C for A in eq.5: B + C + 5×C = 8×C - B + E which becomes B + 6×C = 8×C - B + E In the above equation, subtract 6×C from both sides, and add B to both sides: B + 6×C - 6×C + B = 8×C - B + E - 6×C + B which becomes eq.5a) 2×B = 2×C + E
Hint #10
Substitute 3×C + B for E (from eq.3a) in eq.5a: 2×B = 2×C + 3×C + B which becomes 2×B = 5×C + B Subtract B from each side of the above equation: 2×B - B = 5×C + B - B which makes B = 5×C
Hint #11
Substitute 5×C for B in eq.3a: E = 3×C + 5×C which makes E = 8×C
Solution
Substitute 8×C for A and E, 5×C for B and F, and 4×C for D in eq.1: 8×C + 5×C + C + 4×C + 8×C + 5×C = 31 which simplifies to 31×C = 31 Divide both sides of the above equation by 31: 31×C ÷ 31 = 31 ÷ 31 which means C = 1 making A = E = 8×C = 8 × 1 = 8 B = F = 5×C = 5 × 1 = 5 D = 4×C = 4 × 1 = 4 and ABCDEF = 851485