Puzzle for December 16, 2021 ( )
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
eq.4 may be written as: B + F = A + C – B In the above equation, replace A + C with E + F (from eq.2): B + F = E + F – B Subtract F from both sides, and add B to both sides: B + F – F + B = E + F – B – F + B which makes 2×B = E
Hint #2
In eq.5, replace A + B with C + D – A (from eq.6): D + F = C + D – A + E Subtract C and D from each side of the above equation: D + F – C – D = C + D – A + E – C – D which becomes eq.5a) F – C = –A + E Subtract E and C from both sides of eq.2: E + F – E – C = A + C – E – C which becomes eq.2a) F – C = A – E
Hint #3
In eq.5a, substitute A – E for F – C (from eq.2a): A – E = –A + E Add E and A to both sides of the above equation: A – E + E + A = –A + E + E + A which makes 2×A = 2×E Divide both sides by 2: 2×A ÷ 2 = 2×E ÷ 2 which makes A = E and also makes A = E = 2×B
Hint #4
Substitute A for E in eq.2: A + F = A + C Subtract A from both sides of the above equation: A + F – A = A + C – A which makes F = C
Hint #5
Substitute C for F, and 2×B for E in eq.3: C + C = D + 2×B which becomes 2×C = D + 2×B Subtract 2×B from each side of the equation above: 2×C – 2×B = D + 2×B – 2×B which becomes eq.3a) 2×C – 2×B = D
Hint #6
In eq.6, substitute 2×B for A, and 2×C – 2×B for D (from eq.3a): 2×B + B = C + 2×C – 2×B – 2×B which becomes 3×B = 3×C – 4×B Add 4×B to both sides of the above equation: 3×B + 4×B = 3×C – 4×B + 4×B which becomes 7×B = 3×C Divide both sides by 3: 7×B ÷ 3 = 3×C ÷ 3 which makes 2⅓×B = C and also makes F = C = 2⅓×B
Hint #7
Substitute (2⅓×B) for C in eq.3a: 2×(2⅓×B) – 2×B = D which becomes 4⅔×B – 2×B = D which makes 2⅔×B = D
Solution
Substitute 2×B for A and E, 2⅓×B for C and F, and 2⅔×B for D in eq.1: 2×B + B + 2⅓×B + 2⅔×B + 2×B + 2⅓×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 = E = 2×B = 2 × 3 = 6 C = F = 2⅓×B = 2⅓ × 3 = 7 D = 2⅔×B = 2⅔ × 3 = 8 and ABCDEF = 637867