Puzzle for March 14, 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 to both sides of eq.4: D + E + B = A - B + B which becomes D + E + B = A In eq.6, replace A with D + E + B: D + E + B + F = B + C + D + E Subtract B, D, and E from each side of the above equation: D + E + B + F - B - D - E = B + C + D + E - B - D - E which simplifies to F = C
Hint #2
In eq.1, replace B + C + D + E with A + F (from eq.6): A + A + F + F = 26 which becomes 2×A + 2×F = 26 Divide both sides of the above equation by 2: (2×A + 2×F) ÷ 2 = 26 ÷ 2 which becomes A + F = 13 Subtract F from each side: A + F - F = 13 - F which makes eq.1a) A = 13 - F
Hint #3
eq.4 may be written as: A - B = D + E Add the left and right sides of eq.3 to the left and right sides of the above equation, respectively: A - B + (A + B) = D + E + (C - E) which becomes 2×A = D + C In the above equation, substitute F for C, and (13 - F) for A (from eq.1a): 2×(13 - F) = D + F which is equivalent to 26 - 2×F = D + F Subtract F from both sides: 26 - 2×F - F = D + F - F which becomes eq.4a) 26 - 3×F = D
Hint #4
Substitute 13 - F for A (from eq.1a), and 26 - 3×F for D (from eq.4a) in eq.5: F - B = 13 - F + 26 - 3×F which becomes F - B = 39 - 4×F Subtract F from both sides of the above equation: F - B - F = 39 - 4×F - F which becomes -B = 39 - 5×F Multiply both sides by (-1): -B × (-1) = (39 - 5×F) × (-1) which becomes B = -39 + 5×F which may be written as eq.5a) B = 5×F - 39
Hint #5
Substitute 26 - 3×F for D (from eq.4a), 13 - F for A (from eq.1a), and (5×F - 39) for B (from eq.5a) in eq.4: 26 - 3×F + E = 13 - F - (5×F - 39) which is equivalent to 26 - 3×F + E = 13 - F - 5×F + 39 which becomes 26 - 3×F + E = 52 - 6×F Add (3×F - 26) to both sides of the above equation: 26 - 3×F + E + (3×F - 26) = 52 - 6×F + (3×F - 26) which makes eq.4b) E = 26 - 3×F
Solution
In eq.2, substitute 5×F - 39 for B (from eq.5a), 13 - F for A (from eq.1a), and 26 - 3×F for both D and E (from eq.4a and eq.4b): 5×F - 39 + F = 13 - F + 26 - 3×F + 26 - 3×F which becomes 6×F - 39 = 65 - 7×F Add 7×F + 39 to both sides: 6×F - 39 + 7×F + 39 = 65 - 7×F + 7×F + 39 which simplifies to 13×F = 104 Divide both sides by 13: 13×F ÷ 13 = 104 ÷ 13 which makes F = 8 making A = 13 - F = 13 - 8 = 5 (from eq.1a) B = 5×F - 39 = 5×8 - 39 = 40 - 39 = 1 (from eq.5a) C = F = 8 D = E = 26 - 3×F = 26 - 3×8 = 26 - 24 = 2 (from eq.4a and eq.4b) and ABCDEF = 518228