Puzzle for May 8, 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.
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Hint #1
In eq.3, replace A with E + F (from eq.2): B + F = E + F + D Subtract F from each side of the equation above: B + F – F = E + F + D – F which becomes eq.3a) B = E + D
Hint #2
eq.1 may be written as: A + B + C + F + E + D = 33 In the above equation, substitute A + B for C + F (from eq.4), and B for E + D (from eq.3a): A + B + A + B + B = 33 which becomes 2×A + 3×B = 33 Subtract 3×B from both sides: 2×A + 3×B – 3×B = 33 – 3×B which becomes eq.1a) 2×A = 33 – 3×B
Hint #3
Divide both sides of eq.1a by 2: 2×A ÷ 2 = (33 – 3×B) ÷ 2 which becomes eq.1b) A = 16½ – 1½×B
Hint #4
Check: B = even or odd integer ... If B is an even integer, then: 1½×B = an integer Substituting an integer for 1½×B in eq.1b would yield: A = 16½ – integer which would make A = non-integer Since A must be an integer, then: B ≠ even integer which means B = odd integer
Hint #5
eq.5 may be written as: A = (C + D + F) ÷ 3 Multiply both sides of the equation above by 3: 3 × A = 3 × (C + D + F) ÷ 3 which becomes 3×A = C + D + F which may be written as eq.5a) 3×A = C + F + D
Hint #6
In eq.5a, replace C + F with A + B (from eq.4): 3×A = A + B + D Subtract A from each side of the above equation: 3×A – A = A + B + D – A which becomes eq.4a) 2×A = B + D
Hint #7
In eq.1a, substitute B + D for 2×A (from eq.4a): B + D = 33 – 3×B Subtract B from both sides of the equation above: B + D – B = 33 – 3×B – B which becomes eq.1c) D = 33 – 4×B
Hint #8
To make eq.1c true, check several possible odd number values for B: If B = 9, then D = 33 – 4×9 = 33 – 36 = –3 If B = 7, then D = 33 – 4×7 = 33 – 28 = 5 If B = 5, then D = 33 – 4×5 = 33 – 20 = 13 If B < 5, then D > 13 Since D must be a one-digit non-negative integer, then D = 5 which means B = 7
Hint #9
Substitute 7 for B in eq.1b: A = 16½ – 1½×7 which becomes A = 16½ – 10½ which makes A = 6
Hint #10
Substitute 7 for B, and 5 for D in eq.3a: 7 = E + 5 Subtract 5 from both sides of the above equation: 7 – 5 = E + 5 – 5 which makes 2 = E
Hint #11
Substitute 2 for E, and 6 for A in eq.2: 2 + F = 6 Subtract 2 from both sides of the above equation: 2 + F – 2 = 6 – 2 which makes F = 4
Solution
Substitute 4 for F, 6 for A, and 7 for B in eq.4: C + 4 = 6 + 7 which becomes C + 4 = 13 Subtract 4 from both sides of the above equation: C + 4 – 4 = 13 – 4 which makes C = 9 and ABCDEF = 679524