Puzzle for May 25, 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.3, replace A with B + E (from eq.5): C + E = B + E + B which becomes C + E = 2×B + E Subtract E from both sides of the above equation: C + E – E = 2×B + E – E which makes eq.3a) C = 2×B
Hint #2
In eq.2, replace C with 2×B: D = B + 2×B which makes eq.2a) D = 3×B
Hint #3
In eq.6, substitute B + E for A (from eq.5): B + F = B + E + E which becomes B + F = B + 2×E Subtract B from both sides of the equation above: B + F – B = B + 2×E – B which makes F = 2×E
Hint #4
Substitute 2×E for F, and 3×B for D in eq.4: E + 2×E = B + 3×B which becomes 3×E = 4×B Divide both sides of the equation above by 4: 3×E ÷ 4 = 4×B ÷ 4 which makes ¾×E = B
Hint #5
Substitute (¾×E) for B in eq.3a: C = 2×(¾×E) which makes C = 1½×E
Hint #6
Substitute (¾×E) for B in eq.2a: D = 3×(¾×E) which makes D = 2¼×E
Hint #7
Substitute ¾×E for B in eq.5: A = ¾×E + E which makes A = 1¾×E
Solution
Substitute 1¾×E for A, ¾×E for B, 1½×E for C, 2¼×E for D, and 2×E for F in eq.1: 1¾×E + ¾×E + 1½×E + 2¼×E + E + 2×E = 37 which simplifies to 9¼×E = 37 Divide both sides of the above equation by 9¼: 9¼×E ÷ 9¼ = 37 ÷ 9¼ which means E = 4 making A = 1¾×E = 1¾ × 4 = 7 B = ¾×E = ¾ × 4 = 3 C = 1½×E = 1½ × 4 = 6 D = 2¼×E = 2¼ × 4 = 9 F = 2×E = 2 × 4 = 8 and ABCDEF = 736948