Sales supporting the range:

Sale 1:

Sold for 18 keys

Seller: http://backpack.tf/profiles/76561198107085314/#!/compare/1456905600/1456992000

Buyer: http://backpack.tf/profiles/76561198162477054/#!/compare/1456992000/1457078400

X = 18

Sale 2:

Sold for 18 keys

http://www.tf2outpost.com/trade/27658133 - B/O of 18

Seller: http://backpack.tf/profiles/76561197979591643/#!/compare/1452240000/1452326400

http://backpack.tf/profiles/76561198141518227/#!/compare/1451980800/1452412800

Buyer: http://backpack.tf/profiles/76561198069790291/#!/compare/1452240000/1452326400

This sale confused me for a bit with the scattergun kit, but here is what I have come to:

Hilgyman (Unlabelled) bought the Pro. KS Scattergun kit from Notolerance (Buyer) for 18 keys

Notolerance used the 18 keys to immediately buy the Dapper Topper from Willywoo, which is why the keys weren't scanned in his backpack, and the histories don't match.

X = 18

Sale 3:

Sold for 17 keys

http://www.tf2outpost.com/trade/27754413 - No B/O

Seller: http://backpack.tf/profiles/76561198123269933/#!/compare/1453881600/1454054400

Buyer: http://backpack.tf/profiles/76561197961128885/#!/compare/1454054400/1454140800

X = 17

Sale 4:

Quicksold for 15 keys

http://www.tf2outpost.com/trade/27827939 - B/O of 15

Seller: http://backpack.tf/profiles/76561198085691904/#!/compare/1455004800/1455091200

Buyer: http://backpack.tf/profiles/76561198107085314/#!/compare/1455004800/1455091200

X = 15

Additional Sales:

Sale 5:

Sold for Blizzardy Storm Jumper's Jeepcap

Seller: http://backpack.tf/profiles/76561198094232689/#!/compare/1452931200/1453017600

Buyer: http://backpack.tf/profiles/76561198209626716/#!/compare/1453017600/1453104000

X = 13

Sale 6:

Sold for Orbiting Fire Milkman

http://www.tf2outpost.com/trade/27533484 - B/O of 18

Seller: http://backpack.tf/profiles/76561198069501769/#!/compare/1451030400/1451203200

Buyer: http://backpack.tf/profiles/76561198021324906/#!/compare/1450857600/1451203200

Can be used to update the Milkman

Sale 7:

Sold for Green Confetti Armoured Authority + 4 keys

http://www.tf2outpost.com/trade/27686208 - No B/O

Seller: http://backpack.tf/profiles/76561198021324906/#!/compare/1452758400/1453017600

Buyer: http://backpack.tf/profiles/76561198206754736/#!/compare/1452931200/1453017600

Can be used to update the Authority

Sale 8:

Sold with 13 keys for Peace Sign Demoman's Fro

Cannot find seller's backpack

Buyer: http://backpack.tf/profiles/76561198071473172/#!/compare/1452931200/1453190400

Due to the nature of this sale, it should be used to update the fro.

Sale 9:

Bulk for Duped Purple Energy Killer's Kabuto

Seller: http://backpack.tf/profiles/76561198206754736/#!/compare/1453017600/1453104000

Cannot find buyer's backpack

Sale 10:

Bulk for Stormy Storm Tough Stuff Muffs

http://www.tf2outpost.com/trade/27579923

Seller: http://backpack.tf/profiles/76561198155846118/#!/compare/1452672000/1452758400

Buyer: http://backpack.tf/profiles/76561198094232689/#!/compare/1450166400/1452758400

Can be used to update the Tough Stuff Muffs

Sale 11:

Cannot trace sale. Weak compare link, buyer's backpack cannot be found. Sale might even be outdated, hard to tell with the backpack history having a 3 month gap

http://www.tf2outpost.com/trade/27683285 - B/O of 17

Seller: http://backpack.tf/profiles/76561197990703566/#!/compare/1447747200/1454918400

Sales at:

13 (Too low)

15 (Quicksold)

17 (Pure sale)

18 (Pure sale)

18 (Pure sale)

Going with 17-18