Puzzle for June 7, 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 B to both sides of eq.4: E - B + B = B + F + B which becomes eq.4a) E = 2×B + F In eq.3, replace E with 2×B + F (from eq.4a): D = 2×B + F - F which makes D = 2×B
Hint #2
In eq.2, replace D with 2×B: A = B + 2×B which makes A = 3×B
Hint #3
Subtract F from both sides of eq.4a: E - F = 2×B + F - F which becomes eq.4b) E - F = 2×B Subtract F from both sides of eq.6: C + E - D - F = B + D + F - F which becomes C + E - D - F = B + D which may be written as eq.6a) C + E - F - D = B + D
Hint #4
In eq.6a, substitute 2×B for E - F (from eq.4b), and 2×B for D: C + 2×B - 2×B = B + 2×B which makes C = 3×B
Hint #5
Substitute (2×B + F) for E (from eq.4a), and 3×B for A and C in eq.5: (2×B + F) + F = 3×B + 3×B - (2×B + F) which becomes 2×B + 2×F = 6×B - 2×B - F which becomes 2×B + 2×F = 4×B - F In the above equation, subtract 2×B from both sides, and add F to both sides: 2×B + 2×F - 2×B + F = 4×B - F - 2×B + F which becomes 3×F = 2×B Divide both sides by 3: 3×F ÷ 3 = 2×B ÷ 3 which makes F = ⅔×B
Hint #6
Substitute ⅔×B for F in eq.4a: E = 2×B + ⅔×B which makes E = 2⅔×B
Solution
Substitute 3×B for A and C, 2×B for D, 2⅔×B for E, and ⅔×B for F in eq.1: 3×B + B + 3×B + 2×B + 2⅔×B + ⅔×B = 37 which simplifies to 12⅓×B = 37 Divide both sides of the above equation by 12⅓: 12⅓×B ÷ 12⅓ = 37 ÷ 12⅓ which means B = 3 making A = C = 3×B = 3 × 3 = 9 D = 2×B = 2 × 3 = 6 E = 2⅔×B = 2⅔ × 3 = 8 F = ⅔×B = ⅔ × 3 = 2 and ABCDEF = 939682