Puzzle for September 13, 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
Subtract the left and right sides of eq.3 from the left and right sides of eq.4, respectively: A + B - (B + E) = D - B - (C - B) which becomes A + B - B - E = D - B - C + B which becomes eq.4a) A - E = D - C
Hint #2
Add E and F to both sides of eq.2: C - E + E + F = D - F + E + F which becomes eq.2a) C + F = D + E Add E and C to both sides of eq.4a: A - E + E + C = D - C + E + C which becomes eq.4b) A + C = D + E
Hint #3
In eq.2a, replace D + E with A + C (from eq.4b): C + F = A + C Subtract C from each side of the equation above: C + F - C = A + C - C which makes F = A
Hint #4
eq.5 may be written as: F = (A + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × (A + D + E) ÷ 3 which becomes eq.5a) 3×F = A + D + E
Hint #5
In eq.5a, replace A with F, and D + E with C + F (from eq.2a): 3×F = F + C + F which becomes 3×F = 2×F + C Subtract 2×F from both sides of the equation above: 3×F - 2×F = 2×F + C - 2×F which makes F = C
Hint #6
Substitute F for A and C in eq.6: D ÷ E = (F + F) ÷ F which becomes D ÷ E = 2×F ÷ F which becomes D ÷ E = 2 Multiply both sides of the above equation by E: (D ÷ E) × E = 2 × E which makes D = 2×E
Hint #7
Substitute F for A and C, and 2×E for D in eq.4b: F + F = 2×E + E which becomes 2×F = 3×E Divide both sides of the above equation by 2: 2×F ÷ 2 = 3×E ÷ 2 which makes F = 1½×E and also makes A = C = F = 1½×E
Hint #8
Substitute 1½×E for C in eq.3: B + E = 1½×E - B In the above equation, subtract E from both sides, and add B to both sides: B + E - E + B = 1½×E - B - E + B which makes 2×B = ½×E Divide both sides by 2: 2×B ÷ 2 = ½×E ÷ 2 which makes B = ¼×E
Solution
Substitute 1½×E for A and C and F, ¼×E for B, and 2×E for D in eq.1: 1½×E + ¼×E + 1½×E + 2×E + E + 1½×E = 31 which simplifies to 7¾×E = 31 Divide both sides of the above equation by 7¾: 7¾×E ÷ 7¾ = 31 ÷ 7¾ which means E = 4 making A = C = F = 1½×E = 1½ × 4 = 6 B = ¼×E = ¼ × 4 = 1 D = 2×E = 2 × 4 = 8 and ABCDEF = 616846