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Dr. Atichart KettapunDepartment of MathematicsFaculty of Science, Chiang Mai University
Magic Square (One of the solution)
8 1 6
3 5 7
4 9 2
Add numbers 1‐9 to the 3x3 table with the summation each row, column and diagonal line is equal to 15.
A magic square is a square with numbers such that the summation of each of each row, column and diagonal line is the same.
In a n × n magic square, if the numbers in the table are 1, 2, 3, .., n² we will call it a usual magic square.
What is a Magic Square?Each vertical, horizontal and diagonal
sum is the same. We call that that sum that the Magic Constant.We can also modify the magic square to
be unusual. In Lanna, we found some special magic squares which are not usual magic squares.
What is a Magic Square?
Magic Squares in Many Nations A magic square was found
about 4,000 years ago. The Emperor Yu found a
magic square on the back of a turtle while crossing the Yellow River about 2,200 BC. In the Yih King (อีจ้งิ), this square is called Lo-Shu (เลอ่-ซ)ู.
Magic Squares in Many NationsMagic squares are mathematical works.
Moreover, some nations use them as amulets. For examples, The Hebrews use numbers of miracles for
their amulet which is called Yahweh. Islamic people have been using magic
squares as amulets. Tibetans use magic squares as symbols in
seals.
Magic Squares in Many Nations Hebrew
Amulets showing a magical square in Hebrew. The central character is generally used to indicate the name of God.
Magic Squares in Many Nations Iran
A magical square amulet from Iran
Magic Squares in Many Nations Tibet
A 3x3 Magic square on the ancient seal of the Tibetan people.
Magic Squares in Many Nations Tibet
Mandala - The spiritual symbol in the Hinduism or Buddhism explaining about the universe.
Magic Squares in Many Nations India
Magic Square at the Jain ParsvanathaTemple, Khajuraho, India. The Chautisa Yantra: Since each row, column, diagonal, 2x2 sub‐square, and the corners of each 3x3 and 4x4 square, sums to 34 . Parshvanath temple contains an inscription, 954 AD, by builder Pahil, which refers to Chandella Dhanga as the reigning king.Souce : https://www.pinterest.com/agonzlezgmez /magia
Magic Squares in Many Nations Arabia
Magic Squares in Many Nations Spain
Lanna Magic Squares Example of 3x3 magic square
๒๕๕๐-๐๓-๐๙ -๒๕๕๐-๐๓-๐๙ -พบัยนัตส์าธจุง (พระบญุต่อ อปุลวณฺโณ)วดัศรสีวา่งววัลาย ต. หารแกว้ อ.หางดง จ.เชยีงใหม ่
Lanna Magic Squares (cont.) Example of 4x4 magic square
๒๕๕๑-๐๒-๑๙ -พบัยนัตธ์บุญู ชัน้คร ูว.ท่าทุ่ม ชม (พระไชยวทิย ์ธมฺมรโต)วดัทา่ทุม่ ม.๔ ต.สนัพระเนตร อ.สนัทราย จ.เชยีงใหม่
Lanna Magic Squares (cont.) Example of 5x5 magic square
๒๕๕๐-๐๕-๒๙ -พบัยนัตเ์ทียน วดัเม ืองลงัวดัเมอืงลงั ต.บา้นทอ่ อ.เมอืง จ.เชยีงใหม่
Lanna Magic Squares (cont.) Example of 6x6 magic square
๒๕๕๒-๑๒-๑๖-พิธีกรรม ครบูาอโนไชย ธมมฺจินตามนุี วดัปงสนุกเหนือ ต.เวยีงเหนือ อ.เมอืง จ.ลาํปาง
Lanna Magic Squares (cont.) Example of 9x9 magic square
๒๕๕๐-๑๒-๐๓-พบัยนัตว์ดัธาตคุาํพบัโหรา ยนัต ์วดัธาตุคาํ ต.หายยา อ.เมอืง จ.เชยีงใหม่
Tone Yantra (ยันต์โทน)
From a book about Lanna Yantra
(หนงัสอืยนัตแ์ละคาถาของดเีมอืงเหนือ ของอนิสม ไชยชมพ ูหน้า 130)
This Yantra is for safe travelling.ยนัตล์กูนิเทียวทางบม่ ีอนทรายแล (ยนัตล์กูนี้เดินทางปลอดภยัไม่ม ีอนัตราย)
Tone Yantra (cont.)16 14 18 8
19 7 17 13
10 10 12 14
11 15 9 21
Tone Yantra (cont.)16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 21
Reasons that a numbers in this Lanna Yantra is not correct The inventor intended to write it in this way to
make unwanted people confused. Wrong copying since 1 and 2 are very similar.
16 14 18 8
19 7 17 1310 10 12 1411 15 9 21
Numbers in Lanna Language.
Hora Numbers (ตวัเลขโหรา) are popular numers used in astrology, spell and general writing.
Dharma Numbers (ตวัเลขในธมัม)์ are numbers written in Buddhist scriptures about scriptures and literature.
Thai No. Hora No. Dharma No.
Why is the magic constant equal to 56 ?
Tone Yantra (cont.)Praise of the Buddha Chant or Buddhakhoon Chant
บทสรรเสริญ พระพทุธคณุ (ในบทสวดอติปิิโสฯ)
Tone Yantra (cont.)
อติปิิ โส ภะคะวา อะระหงั สมัมาสมัพทุโธ วชิชาจะระณะสมัปันโน สคุะโต โลกะวทิ ูอะนุตตะโร ปรุสิสะทมัมะสาระถ ิสตัถา เทวะมนุสสานงั พทุโธ ภะคะวาติ
Praise of the Buddha Chant or Buddhakhoon Chant
Since this chant contains 56 syllables, we then named this Yantra as “Buddhakoon 56” (ยันต์พุทธคุณ ๕๖).
ActivityPatterns in Buddhakoon 56 Yantra
16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 21Find mathematical patterns in this Yantra as many as you can.
Tone Yantra (cont.)16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 21
16+14+18+8 = 56
19+7+17+13 = 56
10+20+12+14 = 56
11+15+9+21 = 56
16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 21
Tone Yantra (cont.)
16+19+10+11 = 56
14+7+20+15 = 56
18+17+12+9 = 56
8+13+14+21 = 56
Tone Yantra (cont.)16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 2116+7+12+21 = 56 11+20+17+8 = 56
Tone Yantra (cont.)16 14 18 8
19 7 17 13
10 20 12 14
11 15 9 21
16 - 7 = 9
17 - 8 = 9
20-11 = 9
21-12 = 9
Interesting Questions From Lanna Yantra, do ancient people have a great mathematics ability?
Can we find a way that ancient people used to construct the Buddhakhoon Yantra?
Can we make a make a magic square that the magic constant is 60?
Can we make a make a magic square that the magic constant is 61?
Interesting Questions (cont.) What are magic constants we can create from magic squares?
Can we find interesting mathematical patterns in other Lanna Yantra?
What are similarity and difference between Lanna Yantra and magic squares in other countries?
References www.albrechtdurerblog.com https://www.youtube.com/watch?v=ZLPQMKoAN2I (บทสวด) http://the‐magic‐square.blogspot.com/2010/11/tibetan‐
mandala‐and‐3x3‐magic‐square.html (ทเิบต Mandala) หนงัสอื สนุกคดิคณิตศาสตร ์(The Joy of Mathematics) (ภาพตราประทบัของ
ทเิบต) http://www.bedouinsilver.com/jewelry‐and/bedouin‐
jewelry‐as‐protection/text‐and‐number‐amulets/magical‐squares/ (อหิรา่น)
http://www.dek‐d.com/board/view/2072298/ (เลขโหรา-เลขในธมัม)์