Logic riddles

logicmath

What is the smallest number, that can be expressed as the sum of the cubes of two different sets of numbers?
Hardy-Ramanujan discovered 1729 as a magic number. Why 1729 is a magic number? 10^3 + 9^3 = 1729 and 12^3 + 1^3 = 1729 Taxicab number Ta(2)
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73 votes
logicmath

A women walks into a bank to cash out her check. By mistake the bank teller gives her dollar amount in change, and her cent amount in dollars. On the way home she spends 5 cents, and then suddenly she notices that she has twice the amount of her check. How much was her check amount?
The check was for dollars 31.63. The bank teller gave her dollars 63.31 She spent .05, and then she had dollars 63.26, which is twice the check. Let x be the dolars of the check, and y be the cent. The check was for 100x + y cent He was given 100y + x cent Also 100y + x - 5 = 2(100x + y) Expanding this out and rearranging, we find: 98y = 199x + 5 Which doesn't look like enough information to solve the problem except that x and y must be whole numbers, so: 199x ≡ -5 (mod 98) 98*2*x + 3x ≡ -5 (mod 98) 3x ≡ -5 ≡ 93 (mod 98) This quickly leads to x = 31 and then y = 63 Alternative solution by substitution: 98y = 199x + 5 y = (199x + 5)/98 = 2x + (3x + 5)/98 Since x and y are whole numbers, so must be (3x + 5)/98. Call it z = (3x+5)/98 so 98z = 3x + 5, or 3x = 98z - 5 or x = (98z - 5)/3 or x = 32z-1 + (2z-2)/3. Since everything is a whole number, so must be (2z-2)/3. Call it w = (2z-2)/3, so 3w = 2z-2 so z = (3w+2)/2 or z = w + 1 + w/2. So w/2 must be whole, or w must be even. So try w = 2. Then z = 4. Then x = 129. Then y = 262. if you decrease y by 199 and x by 98, the answer is the same: y = 63 and x = 31.
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57 votes
logicmathclean

Consider the following explanation for why 1=2: 1. Start out Let y = x 2. Multiply through by x xy = x2 3. Subtract y2 from each side xy - y2 = x2 - y2 4. Factor each side y(x-y) = (x+y)(x-y) 5. Divide both sides by (x-y) y = x+y 6. Divide both sides by y y/y = x/y + y/y 7. And so... 1 = x/y + 1 8. Since x=y, x/y = 1 1 = 1 + 1 8. And so... 1 = 2 How is this possible?
Step 5 is invalid, because we are dividing by (x-y), and since x=y, we are thus dividing by 0. This is an invalid mathematical operation (division by 0), and so by not followinng basic mathematical rules, we are able to get strange results like these.
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57 votes