Puzzle for November 15, 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.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
In eq.4, replace A with C + D (from eq.2): C + E = C + D + D Subtract C from both sides of the above equation: C + E – C = C + D + D – C which makes E = 2×D
Hint #2
eq.6 may be written as: 10×A + B = 10×C + D + B Subtract B from both sides of the equation above: 10×A + B – B = 10×C + D + B – B which becomes eq.6a) 10×A = 10×C + D
Hint #3
In eq.6a, substitute (C + D) for A (from eq.2): 10×(C + D) = 10×C + D which is equivalent to 10×C + 10×D = 10×C + D Subtract 10×C and D from both sides of the above equation: 10×C + 10×D – 10×C – D = 10×C + D – 10×C – D which simplifies to 9×D = 0 which means D = 0 and also means E = 2×D = 2×0 = 0
Hint #4
In eq.2, substitute 0 for D: C + 0 = A which becomes C = A
Hint #5
eq.3 may be written as: A + B – F = F – B – A + F Add A, B, and F to both sides of the above equation: A + B – F + A + B + F = F – B – A + F + A + B + F which simplifies to eq.3a) 2×A + 2×B = 3×F
Hint #6
eq.5 may be written as: F – A + F = A – B + C – F Add A and F to each side fo the equation above: F – A + F + A + F = A – B + C – F + A + F which becomes eq.5a) 3×F = 2×A – B + C
Hint #7
Substitute 2×A + 2×B for 3×F (from eq.3a) in eq.5a: 2×A + 2×B = 2×A – B + C In the above equation, subtract 2×A from each side, and add B to each side: 2×A + 2×B – 2×A + B = 2×A – B + C – 2×A + B which makes 3×B = C and also makes eq.5b) A = C = 3×B
Hint #8
Substitute (3×B) for A in eq.3a: 2×(3×B) + 2×B = 3×F which is equivalent to 8×B = 3×F Divide both sides of the above equation by 8: 8×B ÷ 8 = 3×F ÷ 8 which makes B = ⅜×F
Hint #9
Substitute (⅜×F) for B in eq.5b: A = C = 3×(⅜×F) which makes A = C = 1⅛×F
Solution
Substitute 1⅛×F for A and C, ⅜×F for B, and 0 for D and E in eq.1: 1⅛×F + ⅜×F + 1⅛×F + 0 + 0 + F = 29 which simplifies to 3⅝×F = 29 Divide both sides of the equation above by 3⅝: 3⅝×F ÷ 3⅝ = 29 ÷ 3⅝ which means F = 8 making A = C = 1⅛×F = 1⅛ × 8 = 9 D = ⅜×F = ⅜ × 8 = 3 and ABCDEF = 939008