Professor Layton and the Miracle Mask Puzzle 073: Ancient Arithmetic Hints and Answer

Puzzle 073 is Ancient Arithmetic, a 45-picarat Arrange puzzle that Alphonse Dalston gives you in the Briefing Room in Monte d'Or, in Chapter 5. Held at City Hall after his interrogation, Dalston has been passing the time with a stone tablet of arithmetic homework from an ancient settlement. The numbers on it are written in strange symbols, and the bottom-right corner, where the result of the last sum should be, has broken off. You fill the gap by choosing two of five symbol tiles.

Video: Ancient Arithmetic solved. It shows the answer, so skip it if you only want the hints.

This page covers story Puzzle 073 in Professor Layton and the Miracle Mask on Nintendo 3DS (US and UK English), using the North American text. If you're after Daily Puzzle 073 or Puzzle 73 from another Layton game, see Wrong Puzzle 073? below.

Professor Layton and the Miracle Mask Puzzle 073 Ancient Arithmetic: on the top, a golden stone tablet with four rows of sums written in angular symbols with plus, minus and equals signs, its bottom-right corner broken off right after the last equals sign; Dalston, a large man with a mustache, a gray shirt, a red tie and green suspenders, grins on the right in front of a purple patterned backdrop; below, an objective bar reading What's missing?, two empty dark-red spaces side by side and a row of five stone tiles showing a triangle, a square, a K-like shape, a B-like shape and a C-like shape — Illustration of the tablet on the top screen and the five symbol tiles on the touch screen. The game's menu buttons aren't shown.

What Puzzle 073 is

Puzzle 073 at a glance
Name	Ancient Arithmetic (Rock Hard Sums in the UK)
Type	Arrange (drag two symbol tiles into the gap)
Picarats	45
Location	Briefing Room, Monte d'Or
Chapter	Chapter 5
Given by	Alphonse Dalston, after his interrogation
Previous puzzle number	Puzzle 072, A Cup of Tiles (Reception)
Next puzzle number	Puzzle 074, Bunny-Hop Swap 3 (Racetrack)

The question reads: "An elite archaeological team unearthed a stone tablet in the remains of an ancient settlement. According to one excited researcher, it's a near-perfect specimen of arithmetic homework from the era. The strange letters represent numbers, while the operators +, -, and = are the same as the ones we use today! Sadly, the bottom-right corner of the tablet is broken. What should the missing section say?" The objective bar sums it up as "What's missing?"

The tablet holds four sums. The first three are complete: a subtraction, an addition and another subtraction, each with a two-symbol result. The fourth is an addition of two two-symbol numbers, and everything after its equals sign is missing. Only five different symbols appear anywhere on the tablet.

How the tiles work

The touch screen has two empty spaces near the top and a row of five stone tiles below them, one for each symbol on the tablet. From left to right they show a triangle, a square, a K-like shape, a B-like shape and a C-like shape. This page uses those names for the shapes.

Drag a tile into one of the two spaces. Each space holds one tile, and there's only one tile of each symbol.
The two spaces are the missing result, read from left to right like every other number on the tablet.
Only the two spaces count. The game checks which tile is in the left space and which is in the right one. The three tiles you don't use stay in the row and don't matter.
Touch Submit when both spaces are filled. Restart puts all five tiles back in the row.

Puzzle names

Puzzle 073 names
Version	Puzzle name
North America	Ancient Arithmetic
UK	Rock Hard Sums
Japanese	古代文字のナゾ (Kodai Moji no Nazo, roughly "the ancient-letters puzzle")

The UK question and hints also differ in small details of wording and punctuation, such as "near perfect" instead of "near-perfect" and no comma before "and" in lists. The tablet, the tiles and the answer are the same in the US and UK English versions.

Wrong Puzzle 073?

Daily Puzzle 073 is Tile Show 01, from the Whose Tile Is It Anyway? set.
Puzzle 026, Mangled Math, is the other Dalston sum puzzle: young Dalston's torn homework sheet on Pebble Lane in Chapter 2.
Puzzle 041, From the Sidelines, is the earlier puzzle in the same Briefing Room, from Chief Sheffield in Chapter 3.
Puzzle 013, Ancient Arti-facts, is Mr. Collins's excavation chart in Chapter 2, not a tablet of sums.
Puzzle 73 in another Layton game is a different puzzle. If yours doesn't have a broken stone tablet and five symbol tiles, you're in the wrong game.

Where you find it

In Chapter 5, your objective becomes "Head to City Hall!" In the Briefing Room, Layton and Luke sit in on Dalston's interrogation by Chief Sheffield. When it's over, Dalston is still in the room, behind the big front desk in the middle. Talk to him. He asks, "What is it, Layton? Are you gonna try and get me to admit to some lies now, too?" Layton answers, "Not at all, Dalston. I'm quite convinced of your innocence." Then Dalston says, "You still into archaeology, Layton? I've been working on this puzzle while I'm stuck here, but I can't figure it out." Then the puzzle starts.

Dalston keeps the puzzle from the end of the interrogation until your objective becomes "Catch the Masked Gentleman!". The story doesn't need it: it doesn't give a reward. If you miss it, it goes to Nanna Grams (known as Elizabeth in the UK version) at the start of Chapter 7.

The Briefing Room: rows of wooden desks with pink chairs, a big wooden desk at the front in the middle, a film projector on a small table at the front left, a small blackboard on the left wall, window blinds on the side walls, a cabinet and a notice board at the back and hanging ceiling lights. A teal ring labelled Dalston: Puzzle 073 marks a spot behind the big front desk in the middle of the room, and numbered rings mark hint coin spots 1 at the projector table, 2 at a ceiling light at the upper right and 3 at the small blackboard on the left wall — Illustration of the Briefing Room. The teal ring marks where Dalston stands, and the numbers mark where the three hint coins are. The people in the room aren't shown.

The Briefing Room has three hint coins. They're under the projector on the small table at the front left (1), at the ceiling light at the upper right (2) and at the small blackboard on the left wall (3). There's no zoomed-in view in this room. Each coin can be collected only once, so if you picked them up in Chapter 3, they won't be there again.

When you're stuck

The symbols look random. They aren't. The game's own wrong-answer message says there's a logical connection between each character and its number. The hints below lead you to it one step at a time.
You're trying to decode the whole tablet at once. Use the three finished sums to check an idea. A good idea has to work in all three.
Submit says you're wrong. Both spaces count, and so does their order. Read the missing result left to right, like the other numbers on the tablet.
You've lost track of the tiles. Restart puts all five back in the row.
You can't find the puzzle. Talk to Dalston in the Briefing Room after his interrogation in Chapter 5, before your objective becomes "Catch the Masked Gentleman!".

Hints, one at a time

These are the game's own hints, word for word from the North American version and in the same order. Open only the next one you need. In the game, Hints 1 to 3 cost one hint coin each and the Super Hint costs two.

Hint 1: one character's value

The character that looks a bit like a C represents the number 8. With this in mind, can you work out the pattern?

Hint 2: a second character

The character that looks a bit like a B represents the number 7. By the way, there's no relationship between the alphabet and the numbers. It's just a coincidence that these characters look like letters of the alphabet.

Hint 3: the ▲ character

The ▲ character represents the number 3. Look closely at the shapes used to represent 3, 7, and 8. If you spot the pattern, it will be easy to work out what the other characters are.

Super Hint: the pattern

The number of sides a character has tells you what number it represents. Now the puzzle is just a question of arithmetic.

Answer and full solution

The answer below covers the value of every symbol, a check of all four sums, which tile goes in which space and a picture of the solved puzzle. Open it when you're ready.

Show the answer

Answer: 87. Drag the C-like tile into the left space and the B-like tile into the right space, then touch Submit. Each symbol stands for the number of sides it has.

The five stone tiles in a row under the heading Count the sides, each with its number and side count below: the triangle 3, 3 sides; the square 4, 4 sides; the K-like shape 5, 5 sides; the B-like shape 7, 7 sides; the C-like shape 8, 8 sides — Illustration of what each symbol stands for: its number of sides.

The four sums in numbers
Row	Symbols	In numbers
1	square triangle − C-like = triangle K-like	43 − 8 = 35
2	B-like C-like + K-like = C-like triangle	78 + 5 = 83
3	K-like square − B-like = square B-like	54 − 7 = 47
4	triangle square + K-like triangle = ?	34 + 53 = 87

Step by step:

Count the sides of each symbol. The triangle has 3, the square 4, the K-like shape 5, the B-like shape 7 and the C-like shape 8. Those are the numbers they stand for.
Check the finished sums. Row 1 reads 43 − 8 = 35, row 2 reads 78 + 5 = 83 and row 3 reads 54 − 7 = 47. All three work, so the key is right.
Work out the last sum. Row 4 reads 34 + 53, which is 87.
Write 87 with the tiles. 8 is the C-like tile, the last tile in the row, and 7 is the B-like tile, next to it. Put the C-like tile in the left space and the B-like tile in the right space, and touch Submit.

The game confirms it: "Correct! The answer looks a bit like CB and stands for 87, as shown above. The number each character represents is actually the number of sides it has!"

The solved puzzle: on the top, the tablet in front of a blue patterned backdrop, with a small number above every symbol and the last sum finished in red with the C-like and B-like symbols, numbered 8 and 7; Dalston grimaces on the right; below, the C-like tile in the left space and the B-like tile in the right space, with the triangle, square and K-like tiles left in the row — Illustration of the tablet with every symbol's number, and the two tiles in place.

If it still doesn't work:

You put in 78. The tiles are the right ones, but swapped. The C-like tile goes on the left.
You used the K-like tile. That one has five sides, not seven. The B-like tile, with its pointed right side, is the 7.
You counted the notch as one side. Every straight edge counts, including each edge of a notch. The C-like shape's notch adds three edges, which is how it gets to 8.

Get it wrong and the game says, "Too bad. The people of the ancient civilization didn't just create the characters at random. There's a logical connection between each character and the number it represents."

Picarats, hints and hint coins

Ancient Arithmetic shows 45 picarats when it starts. The puzzle can award 45, 41 or 36 picarats. Wrong answers may reduce the picarat value. If you quit, Dalston says, "I'm not saying anything else until my lawyer gets here!" When you come back to try again, he asks, "You gonna solve it or not?"

Hints cost hint coins, not picarats. Hints 1 to 3 cost one hint coin each and the Super Hint costs two, five in total. The Briefing Room's three coins aren't enough for every hint, so you'd need coins saved from elsewhere to open them all. Or use the free hints on this page and save your coins.

Does it unlock anything?

Nothing. Ancient Arithmetic doesn't open a minigame item or anything else. Solving it still adds its picarats to your total and counts toward the puzzles you've solved.

Versions

Nintendo 3DS family: this page covers the 3DS game. You drag the tiles on the touch screen and read the tablet on the top screen.
US and UK English versions: the tablet, the tiles and the answer are the same. The UK name is Rock Hard Sums, and the UK question and hints differ in small details of wording and punctuation.

The scene around the puzzle

Dalston has been arrested as the suspected Masked Gentleman. Layton tells him, "Unfortunately, I haven't been able to convince the police of that...yet." Dalston replies, "Well now, that's quite refreshing to hear. Yes, funny thing about making up your mind; you stop caring about anyone else's."

When it's solved, Dalston says, "Doesn't matter if you believe me or not. I don't see these guys letting me go." Layton answers, "I still plan to prove your innocence, Dalston, but I need more evidence." and asks, "Can you hold on for a while longer?" Dalston: "Doesn't look like I have much choice, eh?" Talk to him again and he says, "This case is paper thin! I can't believe these guys are falling for any of this!"

FAQ

Where is the answer to Puzzle 073?

It's in the Answer and full solution section above, hidden until you open it. If you only want a push, open the hints one at a time first.

What is Puzzle 073 called?

Ancient Arithmetic in North America and Rock Hard Sums in the UK. In Japanese it's 古代文字のナゾ.

Where is Puzzle 073?

In the Briefing Room in Monte d'Or, in Chapter 5. Talk to Dalston after his interrogation, before your objective becomes "Catch the Masked Gentleman!".

I missed Puzzle 073. Can I still get it?

Yes. A missed copy goes to Nanna Grams at the start of Chapter 7.

Do I have to solve Puzzle 073?

No. The story doesn't need it. It doesn't give a reward either, only its picarats.

How many picarats is Puzzle 073 worth?

It shows 45 picarats when it starts. The puzzle can award 45, 41 or 36 picarats.

More Miracle Mask puzzles

For the rest of the game's puzzles, see the Miracle Mask puzzle index.