Puzzle for June 8, 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.
* DE and EF are 2-digit numbers (not D×E or E×F).
Scratchpad
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Hint #1
In eq.5, add D to both sides, and subtract E from both sides: F - D + D - E = C + E + D - E which becomes F - E = C + D In eq.3, replace C + D with F - E: B - E = F - E Add E to both sides of the above equation: B - E + E = F - E + E which makes B = F
Hint #2
In eq.2, replace F with B: E + B = A + B Subtract B from each side of the equation above: E + B - B = A + B - B which means E = A
Hint #3
eq.3 may be re-written as: C + D = B - E Add the left and right sides of the above equation to the left and right sides of eq.4, respectively: C - D + C + D = A + E + B - E which becomes eq.3a) 2×C = A + B
Hint #4
Multiply both sides of eq.5 by 2: 2×(F - D) = 2×(C + E) which is equivalent to 2×F - 2×D = 2×C + 2×E Substitute B for F, A + B for 2×C (from eq.3a), and A for E in the above equation: 2×B - 2×D = A + B + 2×A Add 2×D to both sides, and subtract B from both sides: 2×B - 2×D + 2×D - B = A + B + 2×A + 2×D - B which becomes eq.5a) B = 3×A + 2×D
Hint #5
eq.6 may be written as: 10×D + E - (10×E + F) = B + D which is the same as 10×D + E - 10×E - F = B + D Substitute B for F in the above equation: 10×D + E - 10×E - B = B + D Add B to each side, and subtract D from each side: 10×D + E - 10×E - B + B - D = B + D + B - D which becomes eq.6a) 9×D - 9×E = 2×B
Hint #6
Substitute (3×A + 2×D) for B (from eq.5a), and A for E in eq.6a: 9×D - 9×A = 2×(3×A + 2×D) which is equivalent to 9×D - 9×A = 6×A + 4×D In the above equation, add 9×A to each side, and subtract 4×D from each side: 9×D - 9×A + 9×A - 4×D = 6×A + 4×D + 9×A - 4×D which means 5×D = 15×A Divide both sides by 5: 5×D ÷ 5 = 15×A ÷ 5 which makes D = 3×A
Hint #7
Substitute (3×A) for D in eq.5a: B = 3×A + 2×(3×A) which is equivalent to B = 3×A + 6×A which makes B = 9×A which also makes F = B = 9×A
Hint #8
Substitute 3×A for D, and A for E in eq.4: C - 3×A = A + A Add 3×A to both sides: C - 3×A + 3×A = A + A + 3×A which makes C = 5×A
Solution
Substitute 9×A for B and F, 5×A for C, 3×A for D, and A for E in eq.1: A + 9×A + 5×A + 3×A + A + 9×A = 28 which simplifies to 28×A = 28 Divide both sides by 28: 28×A ÷ 28 = 28 ÷ 28 which means A = 1 making B = F = 9×A = 9 × 1 = 9 C = 5×A = 5 × 1 = 5 D = 3×A = 3 × 1 = 3 E = A = 1 and ABCDEF = 195319