Puzzle for March 30, 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
In eq.4, replace F with B + C (from eq.2): B + B + C = A + C which becomes 2×B + C = A + C Subtract C from each side of the equation above: 2×B + C - C = A + C - C which makes eq.4a) 2×B = A
Hint #2
In eq.5, replace E + F with A + D (from eq.3): A + B = A + D - B In the above equation, subtract A from both sides, and add B to both sides: A + B - A + B = A + D - B - A + B which makes 2×B = D
Hint #3
In eq.3, substitute 2×B for both A and D: E + F = 2×B + 2×B which becomes eq.3a) E + F = 4×B
Hint #4
Substitute 2×B for D in eq.6: C + F = B + 2×B + E which becomes eq.6a) C + F = 3×B + E
Hint #5
Subtract B from each side of eq.2: F - B = C Substitute F - B for C in eq.6a: F - B + F = 3×B + E which becomes 2×F - B = 3×B + E Add B to both sides of the above equation: 2×F - B + B = 3×B + E + B which becomes eq.6b) 2×F = 4×B + E
Hint #6
Substitute E + F for 4×B (from eq.3a) into eq.6b: 2×F = E + F + E which becomes 2×F = 2×E + F Subtract F from each side of the equation above: 2×F - F = 2×E + F - F which makes F = 2×E
Hint #7
Substitute 2×E for F in eq.3a: E + 2×E = 4×B which makes 3×E = 4×B Divide both sides of the above equation by 4: 3×E ÷ 4 = 4×B ÷ 4 which makes ¾×E = B
Hint #8
Substitute ¾×E for B in eq.4a: 2×(¾×E) = A which makes 1½×E = A and also makes 1½×E = A = D
Hint #9
Substitute 2×E for F, and ¾×E for B in eq.2: 2×E = ¾×E + C Subtract ¾×E from both sides of the above equation: 2×E - ¾×E = ¾×E + C - ¾×E which makes 1¼×E = C
Solution
Substitute 1½×E for A and D, ¾×E for B, 1¼×E for C, and 2×E for F in eq.1: 1½×E + ¾×E + 1¼×E + 1½×E + E + 2×E = 32 which simplifies to 8×E = 32 Divide both sides of the above equation by 8: 8×E ÷ 8 = 32 ÷ 8 which means E = 4 making A = D = 1½×E = 1½ × 4 = 6 B = ¾×E = ¾ × 4 = 3 C = 1¼×E = 1¼ × 4 = 5 F = 2×E = 2 × 4 = 8 and ABCDEF = 635648