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    <p>There are several ways of defining what "not biased" means in this case. I assume that what you are looking for is that every possible subtraction problem from the allowed problem space is chosen with equal probability. Quick and dirty approach:</p> <ol> <li>Pick random x in [x_min, x_max]</li> <li>Pick random y in [y_min, y_max]</li> <li>If x - y &lt; answer_min, discard <strong>both x and y</strong> and start over.</li> </ol> <p>Note the bold part. If you discard only y and keep the x, your problems will have an uniform distribution in x, not in the entire problem space. You need to ensure that for every valid x there is at least one valid y - this is not the case for your original choice of ranges, as we'll see later.</p> <p>Now the long, proper approach. First we need to find out the actual size of the problem space.</p> <p>The allowed set of subtrahends is determined by the minuend:</p> <pre><code>x in [21, 99] y in [11, x-10] </code></pre> <p>or using symbolic constants:</p> <pre><code>x in [x_min, x_max] y in [y_min, x - answer_min] </code></pre> <p>We can rewrite that as</p> <pre><code>x in [21, 99] y = 11 + a a in [0, x-21] </code></pre> <p>or again using symbolic constants</p> <pre><code>x in [x_min, x_max] y = y_min + a a in [0, x - (answer_min + y_min)]. </code></pre> <p>From this, we see that valid problems exist only for x >= (answer_min + y_min), and for a given x there are x - (answer_min + y_min) + 1 possible subtrahents.</p> <p>Now we assume that x_max does not impose any further constraints, e.g. that answer_min + y_min >= 0:</p> <pre><code>x in [21, 99], number of problems: (99 - 21 + 1) * (1 + 78+1) / 2 x in [x_min, x_max], number of problems: (x_max - x_min + 1) * (1 + x_max - (answer_min + y_min) + 1) / 2 </code></pre> <p>The above is obtained using the formula for the sum of an arithmetic sequence. Therefore, you need to pick a random number in the range [1, 4740]. To transform this number into a subtraction problem, we need to define a mapping between the problem space and the integers. An example mapping is as follows:</p> <ul> <li>1 &lt;=> x = 21, y = 11</li> <li>2 &lt;=> x = 22, y = 12</li> <li>3 &lt;=> x = 22, y = 11</li> <li>4 &lt;=> x = 23, y = 13</li> <li>5 &lt;=> x = 23, y = 12</li> <li>6 &lt;=> x = 23, y = 11</li> </ul> <p>and so on. Notice that x jumps by 1 when a triangular number is exceeded. To compute x and y from the random number r, find the lowest triangular number t greater than or equal to r, preferably by searching in a precomputed table; write this number as q*(q+1)/2. Then x = x_min + q-1 and y = y_min + t - r.</p> <p>Complete program:</p> <pre><code>import random x_min, x_max = (21, 99) y_min = 11 answer_min = 10 triangles = [ (q*(q+1)/2, q) for q in range(1, x_max-x_min+2) ] upper = (x_max-x_min+1) * (1 + x_max - (answer_min + y_min) + 1) / 2 for i in range(0, 20): r = 1 + random.randrange(0, upper) (t, q) = next(a for a in triangles if a[0] &gt;= r) x = x_min + q - 1 y = y_min + t - r print "%d - %d = ?" % (x, y) </code></pre> <p>Note that for a majority of problems (around 75%), x will be above 60. This is correct, because for low values of the minuend there are fewer allowed values of the subtrahend.</p>
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    1. USKrzysztof Kosiński
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    1. COThanks! I've put in the discard approach for the moment, but I'm interested in understanding the complete way to calculate it, so I'll have a look at the full code sometime later. I was actually *thinking* of uniform distribution of the minuend over its possible values, not caring about the other ones, but on reflection I think this is actually more appropriate.
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