Puzzle for June 3, 2020 ( )
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
In eq.6, replace D (on the left side of the equation only) with C – D + E (from eq.4): B – C + C – D + E – E = A – D which becomes B – D = A – D Add D to both sides of the equation above: B – D + D = A – D + D which makes B = A
Hint #2
In eq.4, replace C with A + D (from eq.5): D = A + D – D + E which becomes eq.4a) D = A + E
Hint #3
In eq.3, substitute A + E for D (from eq.4a): A + B – E = A + E + E which becomes A + B – E = A + 2×E In the equation above, add E to each side, and subtract A from each side: A + B – E + E – A = A + 2×E + E – A which makes B = 3×E and also makes A = B = 3×E
Hint #4
Substitute 3×E for A in eq.4a: D = 3×E + E which makes D = 4×E
Hint #5
Substitute 3×E for A, and 4×E for D in eq.5: C = 3×E + 4×E which makes C = 7×E
Hint #6
Substitute 3×E for both A and B in eq.2: F = 3×E + 3×E + E which makes F = 7×E
Solution
Substitute 3×E for A and B, 7×E for C and F, and 4×E for D in eq.1: 3×E + 3×E + 7×E + 4×E + E + 7×E = 25 which simplifies to 25×E = 25 Divide both sides of the equation above by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = B = 3×E = 3 × 1 = 3 C = F = 7×E = 7 × 1 = 7 D = 4×E = 4 × 1 = 4 and ABCDEF = 337417