Editing Triangular number (section)

==Applications==
{{Anchor|Handshake problem}}<!-- "Handshake problem" redirects here. -->
[[File:handshake_problem_visual_proof.svg|thumb|upright|[[Proof without words]] that the number of possible handshakes between n people is the (n−1)th triangular number]]
The triangular number {{mvar|T<sub>n</sub>}} solves the '''handshake problem''' of counting the number of handshakes if each person in a room with {{math|''n'' + 1}} people shakes hands once with each person. In other words, the solution to the handshake problem of {{mvar|n}} people is {{math|''T''<sub>''n''−1</sub>}}.<ref>{{cite web |url=http://www.mathcircles.org/node/835 |title=The Handshake Problem &#124; National Association of Math Circles |website=MathCircles.org |access-date=12 January 2022 |archive-url=https://web.archive.org/web/20160310182700/http://www.mathcircles.org/node/835 |archive-date=10 March 2016 }}</ref> 

Equivalently, a [[fully connected network]] of {{mvar|n}} computing devices requires the presence of {{math|''T''<sub>''n'' −&thinsp;1</sub>}} cables or other connections.

A triangular number <math>T_{n} </math> is equivalent to the number of principal rotations in dimension <math>n+1</math>. For example, in five dimensions the number of principal rotations is 10 which is <math>T_{4}</math>.<ref>https://henders.one/2022/05/09/lost-4D-rotation/</ref>

In a tournament format that uses a round-robin [[group stage]], the number of matches that need to be played between {{mvar|n}} teams is equal to the triangular number {{math|''T''<sub>''n'' −&thinsp;1</sub>}}. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.

{{central_polygonal_numbers.svg}}
One way of calculating the [[depreciation]] of an asset is the [[depreciation#Sum-of-years-digits method|sum-of-years' digits method]], which involves finding {{mvar|T<sub>n</sub>}}, where {{mvar|n}} is the length in years of the asset's useful life. Each year, the item loses {{math|(''b'' − ''s'') × {{sfrac|''n'' − ''y''|''T<sub>n</sub>''}}}}, where {{mvar|b}} is the item's beginning value (in units of currency), {{mvar|s}} is its final salvage value, {{mvar|n}} is the total number of years the item is usable, and {{mvar|y}} the current year in the depreciation schedule. Under this method, an item with a usable life of {{mvar|n}} = 4 years would lose {{sfrac|4|10}} of its "losable" value in the first year, {{sfrac|3|10}} in the second, {{sfrac|2|10}} in the third, and {{sfrac|1|10}} in the fourth, accumulating a total depreciation of {{sfrac|10|10}} (the whole) of the losable value.

[[Board game]] designers Geoffrey Engelstein and Isaac Shalev describe triangular numbers as having achieved "nearly the status of a mantra or koan among [[Game design|game designers]]", describing them as "deeply intuitive" and "featured in an enormous number of games, [proving] incredibly versatile at providing escalating rewards for larger sets without overly incentivizing specialization to the exclusion of all other strategies".<ref>{{Cite book |last1=Engelstein |first1=Geoffrey |last2=Shalev |first2=Isaac |date=2019-06-25 |title=Building Blocks of Tabletop Game Design |url=http://dx.doi.org/10.1201/9780429430701 |doi=10.1201/9780429430701|isbn=978-0-429-43070-1 |s2cid=198342061 }}</ref>
{| class="wikitable" style="text-align:center; margin:0 auto;"
|+ Relationship between the maximum number of pips on an end of a [[domino]] and the number of dominoes in its set<br />(values in bold are common)
! Max. pips
| 0 || 1 || 2 || 3 || 4 || 5
! 6
| 7 || 8
! 9
| 10 || 11
! 12
| 13 || 14
! 15
| 16 || 17
! 18
| 19 || 20 || 21
|-
! ''n''
| 1 || 2 || 3 || 4 || 5 || 6
! 7
| 8 || 9
! 10
| 11 || 12
! 13
| 14 || 15
! 16
| 17 || 18
! 19
| 20 || 21 || 22
|-
! ''T<sub>n</sub>''
| 1 || 3 || 6 || 10 || 15 || 21
! 28
| 36 || 45
! 55
| 66 || 78
! 91
| 105 || 120
! 136
| 153 || 161
! 190
| 210 || 231 || 253
|}