Brain Teasers
Biggest Number
Using the digits 1 to 9, create two numbers which when multiplied together give you the highest number. For example, 12345678 x 9 = 111111102. Clearly there are higher products. What is the highest?
Hint
One number has 5 digits.Answer
9642 x 87531 makes 843,973,902.Hide Hint Show Hint Hide Answer Show Answer
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Yay first comment! great one!
Good teaser! Any solutions?
Great teaser. Here's my solution.
1) First imagine a slice of z=xy in the plane that satisfies x+y=C, where C is any constant. The result is a parabola that looks like y=xC -x^2 (in a new set of axes, of course). This new function is maximized when its derivative is zero. ie, 0=C-2x, or x=C/2. Back to the original axes, x=C/2 implies y=C/2, so the function is maximized when x and y are equal. Furthermore, the function is increasing until x=y, then it is decreasing. This means that if the difference of x and y is minimized, (keeping x+y constant), the product will be maximized.
2) As we choose different values of C, we see that as C increases, the peak value of z increases as well. This indicates, to some degree, we should choose x and y such that their sum is large.
3) First, to minimize the difference one number (x) will have 5 digits and the other (y) will have 4. Second, to maximize the sum of x and y, x should have 9 in the ten-thousands place. The two numbers should have 8 and 7 in the thousands place, 6 and 5 in the hundreds....
4) Now to minimize the difference between x and y, for each place, y should have the larger of the two numbers. So x = 97531 and y = 8642.
1) First imagine a slice of z=xy in the plane that satisfies x+y=C, where C is any constant. The result is a parabola that looks like y=xC -x^2 (in a new set of axes, of course). This new function is maximized when its derivative is zero. ie, 0=C-2x, or x=C/2. Back to the original axes, x=C/2 implies y=C/2, so the function is maximized when x and y are equal. Furthermore, the function is increasing until x=y, then it is decreasing. This means that if the difference of x and y is minimized, (keeping x+y constant), the product will be maximized.
2) As we choose different values of C, we see that as C increases, the peak value of z increases as well. This indicates, to some degree, we should choose x and y such that their sum is large.
3) First, to minimize the difference one number (x) will have 5 digits and the other (y) will have 4. Second, to maximize the sum of x and y, x should have 9 in the ten-thousands place. The two numbers should have 8 and 7 in the thousands place, 6 and 5 in the hundreds....
4) Now to minimize the difference between x and y, for each place, y should have the larger of the two numbers. So x = 97531 and y = 8642.
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