Wall of Insight

Introduction
'You have all completed your apprenticeships, have served and studied hard for these past seven years. Your studies have been thorough but you, unlike many, have passed and can now no longer be classed as mere apprentices in the Arcane Academy of Meliithra. However, before you can be classed as masters - that most prestigious of titles, respected throughout the land - there is, as you well know, one final step. You must choose which of the five Paths you will follow - and you must qualify for it.

You have all chosen to follow the Path of the Philosopher. Not for you the warrior Path of the Battlemage, fighter for justice, nor the Path of the Healer, worthy though it may be to heal the sick. Equally you have rejected the Path of the Artificer, the tinkerers, who work on devices and techniques, as well as the Path of the Loremaster, who merely delves and catalogues the knowledge of the past. No, you have decided upon the Path of the Philosopher, the highest calling, the way of true enquiry, where you will probe the realms of higher mathematics, ponder the meaning of truth itself and tease out the deepest secrets of nature and the workings of the world.

Only the finest of minds are suited to the Path of the Philosopher and our test represents this. It is simple: one at a time, walk in to the room of testing, yonder. Ahead of you you will see the Wall of Insight. When you have gained access to the room beyond the Wall of Insight, come to me and you will be admitted to the Path of the Philosopher. Be warned though: we do not admit error. Though you may attempt the test at any time you wish, you may only make one error. Should you make a second, you will die. If, after pondering over the Wall, you wish to reconsider your Path, then I wish you luck for your testing for one of the other, lesser, Paths.'
Speech made by Deriilan Gerinii, First Thinker of the Path of the Philosopher at the Arcane Academy of Meliithra to some recently graduated apprentices.

More details
Once an apprentice has graduated, he must gain access to one of the five Paths. These are all accounted of equal worth, though naturally each Path feels their own is superior. Each, to get in, has their own test; some apprentices never succesd in any and remain unMasters for life. Others stay as unMasters for a short time before gaining access to a Path; however, most qualify for a Path immediately after graduating (not always their first choice). The test for the Path of the Philosophy is as described above and below.

The Wall of Insight
On the wall are three things.

1) An 11 by 10 grid of tiles, each with a picture of something on.
2) A small panel with a poem on it.
3) A large 3 by 3 grid, each with writing in it.

If the correct tiles are touched in the right order, the wall swings back and you can pass. The room beyond is bare, but a person gaining in to it will be internally marked by a spell. If they then go report to the First Thinker, he will recognise this mark and admit them to the Path of the Philosopher.

If a wrong tile is touched, you feel a sharp shock. If, at any time in the future, you touch a wrong tile again, you die.

Details:
The tile grid. Starting from the top left and going from left to right and then doing the second row from the top from left to right, etc., the pictures on the tiles are:
i)Rose, Pie, Crown, Walrus, Iris, Diamond, Woman, Hawk, Shield, Fool, Horn
ii)Volcano, Fire, Armour, Fish, Hart, Dolphin, Prince, Star, Sapphire, Bread, Comet
iii)Lamb, Heart, Rain, Ant, Peach, City, Tiger, Orange, Pearl, Noose, Boar
iv) Topaz, Mountain, Bow, Cat, Mouse, Girl, King, Gold, Head, Opal, Elephant
v) Moon, Monkey, Troll, Plum, River, Compass rose, Monk, Fox, Lightning, Whale, Emerald.
vi) Air, Willow, Baby, Scythe, Knight, Axe, Farm, Cloud, Sword, Birch, Mongoose
vii) Apple, Ruby, Abacus, Feet, Centaur, Tulip, Lark, Bee, Ship, Dragon, Sun
viii)Eagle, Island, Aardvark, Cow, Man, Heart, Boy, Queen, Silver, Skull, Snow
ix)Garnet, Plough, Mace, Princess, Lemon, Hand, Serpent, Dog, Bear, Oak, Elm
x)Wind, Ox, Earth, Horse, Throne, Dagger, Lily, Conifer, Water, Swan, Sceptre.

2) The poem on the small panel say:

'The odds you will proceed are less than even;
The following incantation you must speak:
'Monashi Catalan Fibonacci Olmera'
But only those most worthy will it pass.

A three, a four, these aspects are a plane,
A third dimension beckons if you deign.
Yet even if a cube, best not forget:
The prime commandment is factorial yet.'

3) The three by three grid of bits of writing of various length. The segments say:

Top left:
'Boar, king, ruby, willow, star.'

Top centre:
'When patience would first flee the night,
When elements take up the fight,
When fear stifles mortal might,
Three swords must come again.
But when? When?'

Top right:
'Starts, People, Ashari, Animals, Dorrani, Vegetables.'

Middle left:
'Pure east,
Shy south,
Noble north,
Wilful west.
Which symbols will you touch?'

Middle centre:
'Read! Virtue, love and honour be your guide. Which lore will be your saviour? When shall the dead awake and claim their glory? If you would have power and glory, choose so that the second shadow of the sun's last setting obscures the heartland. If you would rule with benevolence and love, choose so that the third shadow of the moon reveals the splendour of paradise. Which will be your choice? Who shall tremble at the passage of your passing? As you must start you must begin and as you would govern, so must you rule. Choose as your heart does sing.'

Middle right:
'The heavy rains show the way one must step. We must obey.'

Bottom left:
'Ardo, Veracity, Regalia, Okandar.'

Bottom centre:
'Right two squares, up to the oldest, first left, third right and then down to the empty square where you began.'

Bottom right:
'Lion is hungry, lamb is fearful
Eagle in eyrie, ox in pasture,
Hart horn-crowned, hawk is swiftest
Swan the whitest, serpent coldest.'

Solution
The cryptic instructions in the verses underneath are all red herrings. The key is the small panel with the poem on it. In the poem 9 number sequences are mentioned in this order:
Odd numbers
Even numbers
Catalan numbers
Fibonacci numbers
Triangle numbers
Square numbers
Cube numbers
Prime numbers
Factorial numbers.

There are also 9 cryptic messages on the tiles.
If you take the first few numbers in each series and apply it to one of the cryptic messages (i.e. if the first numbers are 1,4,9,16,25 then you take the 1st, the 4th, the 9th, the 16th and the 25th letter) then you get a word. You thus have 9 words. If you put them in the order the series were mentioned you get this sentence:
Press in reverse: Aardvark, Sapphire, Throne, Rose, Oak, Whale.
If you press these tiles in reverse order then you will go through.

NB: Of course, if you don't know what Catalan numbers are and press them in the right order and get shocked, you might guess that the missing word is 'reverse' - but do you dare risk it?

Of course, this could be used in any dungeon, not necessarily with the backstory given here.