Puzzle for December 21, 2019 ( )
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 and D to both sides of eq.3: A - B + B + D = B - D + B + D which becomes eq.3a) A + D = 2×B In eq.4, replace A + D with 2×B (from eq.3a): eq.4a) C + F = 2×B
Hint #2
eq.1 may be re-written as: A + D + B + E + C + F = 20 In the equation above, substitute 2×B for A + D (from eq.3a) and for C + F (from eq.4a): 2×B + B + E + 2×B = 20 which becomes 5×B + E = 20 which may be written as eq.1a) 20 = 5×B + E
Hint #3
Subtract C and D from both sides of eq.4: C + F - C - D = A + D - C - D which becomes F - D = A - C In the above equation, replace A - C with B × E (from eq.6): eq.4a) F - D = B × E
Hint #4
In eq.4a, replace F with D + E (from eq.2): D + E - D = B × E which simplifies to E = B × E To make the above equation true, then: B = 1 and / or E = 0
Hint #5
To make eq.1a true, check the two possible values for B and E: If E = 0, then 20 = 5×B + 0 = 5×B which makes B = 20 ÷ 5 = 4 If B = 1, then 20 = 5×1 + E = 5 + E which makes E = 20 - 5 = 15 Since E must be a one-digit integer, then E ≠ 15 which means B ≠ 1 making E = 0 and B = 4
Hint #6
Substitute 0 for E in eq.2: F = D + 0 which means F = D
Hint #7
Substitute D for F in eq.4: C + D = A + D Subtract D from both sides: C + D - D = A + D - D which makes C = A
Hint #8
Substitute 0 for E, D for F, and 4 for B in eq.5: D - 0 + D = 4 + C - D Add D to both sides, and subtract 4 from both sides: D - 0 + D + D - 4 = 4 + C - D + D - 4 which makes 3×D - 4 = C and which also makes eq.5a) A = C = 3×D - 4
Solution
Substitute 3×D - 4 for A and C (from eq.5a), 4 for B, 0 for E, and D for F in eq.1: 3×D - 4 + 4 + 3×D - 4 + D + 0 + D = 20 which simplifies to 8×D - 4 = 20 Add 4 to each side of the above equation: 8×D - 4 + 4 = 20 + 4 which becomes 8×D = 24 Divide both sides by 8: 8×D ÷ 8 = 24 ÷ 8 which means D = 3 making A = C = 3×D - 4 = 3×3 - 4 = 9 - 4 = 5 (from eq.5a) F = D = 3 and ABCDEF = 545303