Lifting straight from that article:
There’s also the Puratto Kodama Economy Plan, which allows you to travel on the Kodama from Tokyo to Kyoto for 10,100 – 12,900 yen one-way, depending on the season. This plan can only be purchased at least a day in advance, so call or visit JR Tokai Tours for booking.
The link to Puratto Kodama Economy Plan: http://www.jrtours.co.jp/kodama/en/
10,100yen, if you travel between October 1 to December 27, 2016.
There are these discount ticket shops as well, but they are probably inferior to Puratto Kodama Economy Plan
Note the file photo: 12,600yen to Kyoto.
Opting to skip the reserved seat and go for the unreserved seat will save you between 510yen (for Hikari, slower shinkansen) to 830yen (for Nozomi, faster Shinkansen).
Only applies to regular tickets, about 13,600yen for Hikari and 13,910yen for Nozomi, price accurate for October 15, 2016.