| <p> | |
| It's dinner time! | |
| A group of <strong>N</strong> Foxen are standing silently in a field, which can be represented as an infinite number line, | |
| patiently waiting for their meals to make an appearance. | |
| The <em>i</em>th Fox is standing at position <strong>P<sub>i</sub></strong>, | |
| with no two Foxen standing at the same position. | |
| There's also one hole in the ground at each each integral position on the number line. | |
| Each of these holes is the entrance to a mole's den, and the Foxen know that some of these delicious critters are | |
| bound to show up sooner or later! | |
| </p> | |
| <p> | |
| A little-known fact about Foxen is that, in addition to having an acute array of regular senses, | |
| they possess a SONAR-like ability to emit imperceptible sound waves and use them to discern objects at great distances. | |
| The <em>i</em>th Fox has tuned their wavelength to a distance of <strong>R<sub>i</sub></strong>, | |
| allowing them to only detect moles which emerge from holes at a distance of exactly <strong>R<sub>i</sub></strong> away from them | |
| (that is, at either position <strong>P<sub>i</sub></strong> - <strong>R<sub>i</sub></strong> or | |
| <strong>P<sub>i</sub></strong> + <strong>R<sub>i</sub></strong>). | |
| </p> | |
| <p> | |
| All of a sudden, some number of moles have just popped up from various holes all at once! | |
| No mole popped up at any Fox's position, no two moles popped up from the same hole, and | |
| every mole was detected by at least one Fox. | |
| Furthermore, each Fox <em>i</em> has determined that there's <i>exactly</i> 1 mole | |
| at a distance of <strong>R<sub>i</sub></strong> away from it (as opposed to there being either 0 or 2 such moles). | |
| </p> | |
| <p> | |
| Following this initial event, there's been quite some commotion. | |
| Some moles may have retreated back underground, and some new moles may have emerged, all in any order. | |
| At every point in time, the set of moles on the surface is subject to all of the same restrictions as before, with one difference: | |
| Each Fox <em>i</em> continues to be sure that <i>at least</i> 1 mole is still present | |
| at a distance of <strong>R<sub>i</sub></strong> away from it, but can no longer determine whether or not | |
| there are perhaps now 2 such moles instead. | |
| </p> | |
| <p> | |
| After some time of this, the Foxen have decided that they're ready to pounce and "invite" some of the moles | |
| currently on the surface over for dinner. | |
| Unfortunately, they've started to become rather overwhelmed with trying to keep track of which moles | |
| may be on the surface, or even roughly how many of them there might be. | |
| Assuming that the Foxen's initial observations were correct, and that some unknown amount of time has since gone by | |
| with moles surfacing or departing, please help the Foxen determine the number of different quantities of moles which | |
| could possibly have ended up on the surface. | |
| </p> | |
| <p> | |
| If it's impossible for their set of initial observations to have been accurate in the first place, output -1 instead. | |
| </p> | |
| <h3>Input</h3> | |
| <p> | |
| Input begins with an integer <strong>T</strong>, the number of different fields. | |
| For each field, there is first a line containing the integer <strong>N</strong>. | |
| Then <strong>N</strong> lines follow, the <em>i</em>th of which contains the space-separated integers | |
| <strong>P<sub>i</sub></strong> and <strong>R<sub>i</sub></strong>. | |
| </p> | |
| <h3>Output</h3> | |
| <p> | |
| For the <em>i</em>th field, print a line containing "Case #<strong>i</strong>: " | |
| followed by a single integer, the number of different quantities of moles which could possibly end up on the surface at any point, | |
| or -1 if the Foxen's initial observations must have been inaccurate. | |
| </p> | |
| <h3>Constraints</h3> | |
| <p> | |
| 1 ≤ <strong>T</strong> ≤ 30<br /> | |
| 1 ≤ <strong>N</strong> ≤ 5,000 <br /> | |
| 0 ≤ <strong>P<sub>i</sub></strong> ≤ 1,000,000,000 <br /> | |
| 1 ≤ <strong>R<sub>i</sub></strong> ≤ 1,000,000,000 <br /> | |
| </p> | |
| <h3>Explanation of Sample</h3> | |
| <p> | |
| In the first case, it's possible for there to eventually be 1 mole (at either position -1 or 1), or 2 moles (at both positions -1 and 1). There can't be 0 moles due to the restriction that the Fox must detect at least 1 of them, and there can't be more than 2 moles as they'd have to be at positions which the Fox is unable to detect. | |
| </p> | |
| <p> | |
| In the third case, it's impossible for a set of moles to have initially popped up such that each Fox would have detected <em>exactly</em> one of them. | |
| </p> | |