Math puzzle worksheet featuring a pyramid of hexagons where students must determine the addition rule to find missing numbers, with a handwritten note at the top.

Let’s solve this step by step.

We have a pyramid of hexagons. The rule (written at the top) says:
“the two hexagon up to the hexagon” — which likely means:
→ Each hexagon is the sum of the two hexagons directly below it.

Let’s test that with what we know.

Look at the bottom row:
- Leftmost: 12,345
- Next: 35,789
- Then: ? (this one is blank and has a gray bar under it — probably the one we need to find)
- Rightmost: 24,608

Now look at the row above:
- First hexagon (left): should be 12,345 + 35,789 = let’s calculate:

12,345 + 35,789 =
→ 12,000 + 35,000 = 47,000
→ 345 + 789 = 1,134
→ Total = 48,134 → but wait! In the diagram, the hexagon above 12,345 and 35,789 is actually labeled 20,451? That doesn’t match.

Wait — maybe I misread the structure.

Actually, looking again:

The hexagon labeled 20,451 is in the middle of the third row from the bottom. It sits above two hexagons: one is 35,789 (bottom row), and the other is the unknown one (with the gray bar). So:

If the rule is “each hexagon = sum of the two below it”, then:

20,451 = 35,789 + ?

But that would mean ? = 20,451 - 35,789 = negative number → impossible for whole numbers.

So maybe the rule is reversed? Maybe each hexagon is the sum of the two *above* it? No, that doesn’t make sense either because the top is biggest.

Wait — let’s check another pair.

Look at the right side:

Bottom row: ? (gray bar) and 24,608 → they feed into the hexagon above them, which is labeled 48,093.

So if 48,093 = ? + 24,608 → then ? = 48,093 - 24,608 = 23,485

Let me calculate that:

48,093
-24,608
= ?

93 - 08 = 85
09 - 60 → can’t do, borrow → 109 - 60 = 49, but since we borrowed, the thousands digit becomes 7 (from 8) → wait, better to do straight subtraction:

48,093
-24,608
________
Start from right:

3 - 8 → borrow → 13 - 8 = 5
8 (became 7 after borrow) - 0 = 7? Wait no:

Actually:

Units: 3 < 8 → borrow from tens → tens is 9, becomes 8, units become 13 → 13 - 8 = 5
Tens: 8 (after borrow) - 0 = 8
Hundreds: 0 < 6 → borrow from thousands → thousands is 8, becomes 7, hundreds become 10 → 10 - 6 = 4
Thousands: 7 (after borrow) - 4 = 3
Ten-thousands: 4 - 2 = 2

Wait, that gives 23,485? Let me write it properly:

48,093
- 24,608
__________
23,485 ← yes!

So the missing hexagon (gray bar) must be 23,485, because 23,485 + 24,608 = 48,093.

Now let’s verify with the left side.

On the left, we have 12,345 and 35,789 feeding into a hexagon above them. What should that be?

12,345 + 35,789 = ?

12,345
+35,789
________
5+9=14 → write 4, carry 1
4+8+1=13 → write 3, carry 1
3+7+1=11 → write 1, carry 1
2+5+1=8
1+3=4
→ 48,134

But in the diagram, the hexagon above 12,345 and 35,789 is not shown — instead, the next level up has 76,426 on the left.

Wait — let’s map the levels clearly.

Label rows from bottom to top:

Row 1 (bottom):
A = 12,345
B = 35,789
C = ? (gray bar) ← this is what we’re solving for
D = 24,608

Row 2:
E = A + B = 12,345 + 35,789 = 48,134
F = B + C = 35,789 + C
G = C + D = C + 24,608 = 48,093 → so C = 48,093 - 24,608 = 23,485 ✔

Then F = 35,789 + 23,485 = ?

35,789
+23,485
________
9+5=14 → 4, carry 1
8+8+1=17 → 7, carry 1
7+4+1=12 → 2, carry 1
5+3+1=9
3+2=5
→ 59,274

But in the diagram, Row 2 has three hexagons: left is empty, middle is 20,451, right is 48,093.

Wait — that doesn’t match. According to our calculation, F should be 59,274, but the diagram shows 20,451 in the middle of Row 2.

That suggests my assumption about the structure might be wrong.

Alternative idea: Maybe the hexagons are arranged such that each upper hexagon is supported by two lower ones, but not necessarily adjacent in the way I thought.

Looking back at the image description:

Top: 100,000

Below it: left is 76,426, right is blank

Below that: left blank, middle 20,451, right 48,093

Bottom: 12,345, 35,789, [gray], 24,608

Perhaps the connections are:

- 100,000 = 76,426 + (blank right)

- 76,426 = (left blank in row 3) + 20,451

- 20,451 = 35,789 + (gray bar)

- 48,093 = (gray bar) + 24,608

Ah! This makes more sense.

So let’s use that.

From 48,093 = (gray bar) + 24,608 → gray bar = 48,093 - 24,608 = 23,485

Also, from 20,451 = 35,789 + (gray bar) → but 35,789 + 23,485 = 59,274 ≠ 20,451 → contradiction.

Unless... maybe it's subtraction? Or different operation?

Wait — perhaps the rule is: each hexagon is the sum of the two below it, but the diagram is drawn with the top being the total, and we work down.

Another approach: start from the top.

Top: 100,000

It is made from two below: 76,426 and X → so X = 100,000 - 76,426 = 23,574

So the blank hexagon to the right of 76,426 is 23,574

Now, 76,426 is made from two below: let's call them Y and 20,451 → so Y = 76,426 - 20,451 = 55,975

Similarly, 23,574 (which we just found) is made from 20,451 and 48,093? But 20,451 + 48,093 = 68,544 ≠ 23,574 → no.

Perhaps 23,574 is made from 48,093 and Z? Doesn't fit.

Let’s list all known values with positions.

Assume the pyramid has 4 rows:

Row 4 (top): H1 = 100,000

Row 3: H2 = 76,426, H3 = ?

Row 2: H4 = ?, H5 = 20,451, H6 = 48,093

Row 1 (bottom): H7 = 12,345, H8 = 35,789, H9 = ? (gray), H10 = 24,608

Standard honeycomb addition: each cell is sum of two cells below it.

So:

H1 = H2 + H3 → 100,000 = 76,426 + H3 → H3 = 23,574

H2 = H4 + H5 → 76,426 = H4 + 20,451 → H4 = 55,975

H3 = H5 + H6 → 23,574 = 20,451 + 48,093? 20,451 + 48,093 = 68,544 ≠ 23,574 → not possible.

Unless H3 is not H5 + H6, but rather H6 and something else.

Perhaps the connectivity is:

- H1 depends on H2 and H3

- H2 depends on H4 and H5

- H3 depends on H5 and H6

- H4 depends on H7 and H8

- H5 depends on H8 and H9

- H6 depends on H9 and H10

Yes! That makes sense for a hexagonal grid.

So:

H5 = H8 + H9 → 20,451 = 35,789 + H9 → H9 = 20,451 - 35,789 = negative → impossible.

Still problem.

Unless the rule is subtraction? But the chapter is "Working with Whole Numbers", and usually these are addition pyramids.

Another possibility: maybe the numbers are placed differently.

Let’s try to use the given numbers to find consistency.

Notice that 12,345 + 35,789 = 48,134

And 35,789 + 24,608 = 60,397

Not matching any.

What if we consider that the gray bar hexagon is part of multiple sums.

From the right side: the hexagon above the gray bar and 24,608 is 48,093, so gray bar = 48,093 - 24,608 = 23,485

From the left side: the hexagon above 12,345 and 35,789 should be 12,345 + 35,789 = 48,134

Then, the hexagon above 48,134 and 20,451 should be 48,134 + 20,451 = 68,585

But in the diagram, the left hexagon in row 3 is 76,426, which is not 68,585.

76,426 - 68,585 = 7,841 — not zero.

Perhaps the middle hexagon in row 2 is not 20,451 for the left path.

Let’s calculate what the top should be if we go left path.

Bottom: 12,345 and 35,789 -> sum 48,134

Then 48,134 and say X -> sum Y

Then Y and Z -> 100,000

But we have 76,426 on the left of row 3.

Assume that 76,426 is the sum of the two below it: which are the left-most of row 2 and the middle of row 2.

Left-most of row 2 is sum of 12,345 and 35,789 = 48,134

Middle of row 2 is 20,451

So 48,134 + 20,451 = 68,585, but 76,426 is given, so difference is 76,426 - 68,585 = 7,841

Not helpful.

Another idea: perhaps the rule is that each hexagon is the sum of the two below it, but the gray bar is not in the position I think.

Let’s look at the handwritten rule: "the two hexagon up to the hexagon" — perhaps it's poorly written, and it means "each hexagon is the sum of the two hexagons below it".

Given that, and the only consistent calculation is from the right side: 48,093 = gray bar + 24,608, so gray bar = 23,485

And for the left side, 12,345 + 35,789 = 48,134, which should be the hexagon above them, but in the diagram, that hexagon is not labeled; instead, the next level has 76,426 on the left, which might be the sum of 48,134 and 20,451, but 48,134 + 20,451 = 68,585 ≠ 76,426.

Unless 20,451 is not the middle of row 2 for the left path.

Perhaps the 20,451 is the sum of 35,789 and the gray bar.

So 20,451 = 35,789 + gray bar → gray bar = 20,451 - 35,789 = -15,338 — impossible.

This is confusing.

Let’s try to use the top number.

100,000 = 76,426 + X, so X = 23,574

Then 76,426 = A + 20,451, so A = 55,975

23,574 = 20,451 + B, so B = 3,123

Then A = 55,975 = 12,345 + 35,789 = 48,134 — not equal.

55,975 - 48,134 = 7,841 again.

Perhaps there's a mistake in the problem or my understanding.

Another thought: maybe the gray bar is not a hexagon to be filled, but an indicator, and we need to find the rule first.

The user wrote at the top: "Rule: the two hexagon up to the hexagon" — perhaps it's "each hexagon is the sum of the two hexagons below it".

Given that, and the only logical place to start is from the bottom right.

We have 48,093 = C + 24,608, so C = 23,485

Then, the hexagon above C and 35,789 is 20,451? But 35,789 + 23,485 = 59,274 ≠ 20,451

Unless 20,451 is not above them.

Perhaps the 20,451 is above 35,789 and the gray bar, but in the diagram, it's positioned between.

Let’s assume that the hexagon labeled 20,451 is indeed the sum of the two below it: 35,789 and the gray bar.

Then 20,451 = 35,789 + G → G = 20,451 - 35,789 = -15,338 — not possible.

So perhaps the rule is subtraction: each hexagon is the difference of the two below it.

For example, 48,093 = |C - 24,608| or something.

But 48,093 - 24,608 = 23,485, as before.

Or 24,608 - C = 48,093 — impossible.

Another idea: perhaps "up to" means that the two below add up to the one above, but the gray bar is not in the bottom row for that purpose.

Let’s count the hexagons.

In a standard pyramid with 4 rows, there are 1+2+3+4 = 10 hexagons.

In the diagram, we have:

- Top: 1 (100,000)

- Row 2: 2 (76,426 and blank)

- Row 3: 3 (blank, 20,451, 48,093)

- Row 4: 4 (12,345, 35,789, gray, 24,608)

So 10 hexagons, good.

Now, typically, each hexagon in row n is supported by two in row n+1.

Specifically:

- The top hexagon (row 1) is supported by the two in row 2.

- Each in row 2 is supported by two in row 3.

- Each in row 3 is supported by two in row 4.

So:

Let me denote:

R1: A = 100,000

R2: B = 76,426, C = ?

R3: D = ?, E = 20,451, F = 48,093

R4: G = 12,345, H = 35,789, I = ? (gray), J = 24,608

Then:

A = B + C => 100,000 = 76,426 + C => C = 23,574

B = D + E => 76,426 = D + 20,451 => D = 55,975

C = E + F => 23,574 = 20,451 + 48,093? 20,451 + 48,093 = 68,544 ≠ 23,574 — still not.

Unless C = F + something else.

Perhaps C is supported by F and J or something.

Standard connectivity for hexagonal grid:

Usually, for a hexagon in row r, position p, it is supported by row r+1, positions p and p+1.

So for R3, position 1 (D) is supported by R4 pos 1 and 2: G and H

R3 pos 2 (E) is supported by R4 pos 2 and 3: H and I

R3 pos 3 (F) is supported by R4 pos 3 and 4: I and J

Then R2 pos 1 (B) is supported by R3 pos 1 and 2: D and E

R2 pos 2 (C) is supported by R3 pos 2 and 3: E and F

R1 pos 1 (A) is supported by R2 pos 1 and 2: B and C

Perfect.

So now we can write equations:

From R4 to R3:

D = G + H = 12,345 + 35,789 = 48,134

E = H + I = 35,789 + I

F = I + J = I + 24,608

From R3 to R2:

B = D + E = 48,134 + E

C = E + F

From R2 to R1:

A = B + C = 100,000

Also, from the diagram, B = 76,426, E = 20,451, F = 48,093

Oh! We have E = 20,451 and F = 48,093 given.

So from E = H + I = 35,789 + I = 20,451 → I = 20,451 - 35,789 = -15,338 — still negative.

But if E = 20,451 is given, and H = 35,789, then I must be negative, which is not allowed for whole numbers.

Unless the rule is not addition, but something else.

Perhaps the rule is that the upper hexagon is the sum, but the numbers are placed in a different order.

Another possibility: maybe " the two hexagon up to the hexagon" means that the two below are added to get the one above, but in this case, for E = 20,451, it is the sum of H and I, so 20,451 = 35,789 + I, which requires I negative.

Perhaps it's the absolute difference or min/max, but unlikely.

Let’s check the numbers again.

Perhaps the 20,451 is not E, but D or something.

In the diagram, the 20,451 is in the middle of the third row from the bottom, which should be E.

But let's calculate what I should be from F.

F = 48,093 = I + J = I + 24,608 → I = 48,093 - 24,608 = 23,485

Then E = H + I = 35,789 + 23,485 = 59,274

But in the diagram, E is labeled as 20,451, which is not 59,274.

So conflict.

Unless the label 20,451 is for a different hexagon.

Perhaps the 20,451 is for D, the left one in row 3.

Let's assume that.

Suppose in row 3, left is 20,451, middle is blank, right is 48,093.

But in the user's description, it's "20,451" in the middle.

Perhaps there's a typo in the problem or in my reading.

Another idea: perhaps the rule is that each hexagon is the sum of the two above it, but that would make the bottom larger, which is not the case.

Let's try to ignore the given labels for a moment and use the top.

A = 100,000 = B + C

B = 76,426, so C = 23,574

B = D + E

C = E + F

Also, D = G + H = 12,345 + 35,789 = 48,134

F = I + J = I + 24,608

E = H + I = 35,789 + I

From B = D + E = 48,134 + E = 76,426 → E = 76,426 - 48,134 = 28,292

Then from E = 35,789 + I = 28,292 → I = 28,292 - 35,789 = -7,497 — still negative.

Same issue.

From C = E + F = 23,574

And F = I + 24,608

E = 35,789 + I

So C = (35,789 + I) + (I + 24,608) = 60,397 + 2I = 23,574

Then 2I = 23,574 - 60,397 = -36,823 → I = -18,411.5 — worse.

So addition doesn't work with the given numbers.

Perhaps the rule is multiplication or other, but unlikely for whole numbers chapter.

Another thought: maybe " the two hexagon up to the hexagon" means that the two below are subtracted or something.

Let’s look at the numbers: 12,345, 35,789, etc.

Notice that 35,789 - 12,345 = 23,444

Close to 23,485? Not really.

48,093 - 24,608 = 23,485

20,451 - 35,789 = -15,338

Not helping.

Perhaps the gray bar is 23,485, and we accept that the 20,451 is a distractor or error, but that seems unlikely.

Let’s calculate what the top should be if we use the bottom numbers with addition.

G=12,345, H=35,789, I=23,485, J=24,608

Then D = G+H = 48,134

E = H+I = 35,789 + 23,485 = 59,274

F = I+J = 23,485 + 24,608 = 48,093 — matches the given F=48,093

Then B = D+E = 48,134 + 59,274 = 107,408

C = E+F = 59,274 + 48,093 = 107,367

Then A = B+C = 107,408 + 107,367 = 214,775 — not 100,000.

But in the diagram, B is given as 76,426, which is not 107,408.

So not matching.

Perhaps the rule is that each hexagon is the average or something, but not for whole numbers.

Let’s try subtraction: suppose each upper hexagon is the difference of the two below it.

For example, F = |I - J| = |I - 24,608| = 48,093

So |I - 24,608| = 48,093 → I - 24,608 = 48,093 or I - 24,608 = -48,093

First: I = 48,093 + 24,608 = 72,701

Second: I = 24,608 - 48,093 = -23,485 — invalid.

So I = 72,701

Then E = |H - I| = |35,789 - 72,701| = 36,912

But in diagram, E = 20,451 — not match.

If E = |I - H| = same thing.

Not working.

Perhaps it's the sum, but the given 20,451 is for a different purpose.

Let’s notice that 76,426 + 23,574 = 100,000, and 23,574 is C.

Then if C = E + F, and F = 48,093, then E = 23,574 - 48,093 = -24,519 — invalid.

I think there might be a mistake in the problem or in the initial assumption.

Perhaps the "20,451" is not in row 3, but in row 2 or something.

Another idea: maybe the hexagon with 20,451 is the one that is sum of 35,789 and the gray bar, but in the diagram, it's written as 20,451, but perhaps it's 59,274 or something.

But the user has it as 20,451.

Let’s calculate 35,789 + 24,608 = 60,397

12,345 + 24,608 = 36,953

Not helping.

Perhaps the rule is that the upper hexagon is the sum, but the gray bar is to be found from the top down.

Start from top: 100,000 = 76,426 + X, so X = 23,574

Then 76,426 = Y + 20,451, so Y = 55,975

23,574 = 20,451 + Z, so Z = 3,123

Then Y = 55,975 = 12,345 + 35,789 = 48,134 — not equal, difference 7,841

Z = 3,123 = I + 24,608? Then I = 3,123 - 24,608 = -21,485 — invalid.

Or Z = |I - 24,608| = 3,123, so I = 24,608 ± 3,123 = 27,731 or 21,485

Then from Y = 55,975 = D + E = (12,345 + 35,789) + (35,789 + I) = 48,134 + 35,789 + I = 83,923 + I

So 83,923 + I = 55,975 → I = 55,975 - 83,923 = -27,948 — not matching.

I think I need to accept that from the right side, with F = 48,093 = I + J = I + 24,608, so I = 23,485, and perhaps the 20,451 is a red herring or there's a typo, but that's not satisfactory.

Let’s look back at the user's handwritten rule: " the two hexagon up to the hexagon" — perhaps it's " the two hexagons below add up to the hexagon above".

And in the diagram, for the hexagon with 48,093, it is above the gray bar and 24,608, so gray bar = 48,093 - 24,608 = 23,485

For the hexagon with 20,451, it is above 35,789 and the gray bar, so 20,451 = 35,789 + 23,485 = 59,274 — but 59,274 ≠ 20,451, so perhaps the 20,451 is incorrect, or perhaps it's 59,274 and miswritten.

Maybe it's 20,451 for a different hexagon.

Another possibility: perhaps the 20,451 is the value for the hexagon that is sum of 12,345 and 35,789, but 12,345 + 35,789 = 48,134, not 20,451.

48,134 - 20,451 = 27,683 — not zero.

Let’s calculate 35,789 - 12,345 = 23,444

Close to 23,485? 23,485 - 23,444 = 41 — not very close.

Perhaps it's the product or other, but unlikely.

Let’s try to see if 100,000 can be reached.

Suppose the gray bar is X.

Then from right: 48,093 = X + 24,608 → X = 23,485

From left: the hexagon above 12,345 and 35,789 is 12,345 + 35,789 = 48,134

Then the hexagon above 48,134 and 20,451 is 48,134 + 20,451 = 68,585

Then the top is 68,585 + (hexagon above 20,451 and 48,093) = 68,585 + (20,451 + 48,093) = 68,585 + 68,544 = 137,129 — not 100,000.

If the hexagon above 20,451 and 48,093 is C, and 100,000 = 76,426 + C, so C = 23,574, then 20,451 + 48,093 = 68,544 ≠ 23,574.

I think the only consistent thing is that for the right side, X = 48,093 - 24,608 = 23,485

And perhaps the 20,451 is meant to be 59,274, but it's written as 20,451 by mistake.

Maybe in the diagram, the 20,451 is for the left hexagon in row 3, not the middle.

Let’s assume that.

Suppose in row 3, left is 20,451, middle is blank, right is 48,093.

Then D = 20,451 = G + H = 12,345 + 35,789 = 48,134 — not match.

20,451 = 12,345 + 35,789? 12,345 + 35,789 = 48,134 ≠ 20,451.

Perhaps 20,451 = |35,789 - 12,345| = 23,444 — close to 23,485? 23,485 - 23,444 = 41, not exact.

23,444 vs 23,485 — difference of 41, not zero.

Perhaps it's 35,789 - 12,345 = 23,444, and the gray bar is 23,485, so not the same.

I recall that in some puzzles, the rule might be that the upper is the sum, but here it's not working.

Let’s calculate the sum of all bottom numbers: 12,345 + 35,789 + X + 24,608 = 72,742 + X

Then if each level sums to twice the previous or something, but complicated.

Perhaps the rule is correct, and we need to find X such that the top is 100,000.

From earlier:

D = 12,345 + 35,789 = 48,134

E = 35,789 + X

F = X + 24,608

B = D + E = 48,134 + 35,789 + X = 83,923 + X

C = E + F = (35,789 + X) + (X + 24,608) = 60,397 + 2X

A = B + C = (83,923 + X) + (60,397 + 2X) = 144,320 + 3X = 100,000

So 144,320 + 3X = 100,000

3X = 100,000 - 144,320 = -44,320

X = -14,773.333 — not integer, and negative.

So impossible with addition.

Therefore, the rule must be different.

Perhaps " the two hexagon up to the hexagon" means that the two below are used to make the one above by subtraction or other operation.

Let’s try: suppose each upper hexagon is the difference of the two below it, with the larger minus smaller.

For example, for F = 48,093 = |X - 24,608|

As before, X = 48,093 + 24,608 = 72,701 or X = 24,608 - 48,093 = -23,485 — take 72,701

Then for E = 20,451 = |35,789 - X| = |35,789 - 72,701| = 36,912 ≠ 20,451

Not match.

If E = |X - 35,789| = 20,451, then X - 35,789 = 20,451 or X - 35,789 = -20,451

So X = 56,240 or X = 15,338

Then for F = |X - 24,608| = 48,093

If X = 56,240, |56,240 - 24,608| = 31,632 ≠ 48,093

If X = 15,338, |15,338 - 24,608| = 9,270 ≠ 48,093

Not match.

Perhaps it's the sum for some, difference for others, but unlikely.

Another idea: perhaps the rule is that the upper hexagon is the sum, but the numbers are in a different base, but that's advanced.

Or perhaps there's a constant added, but not specified.

Let’s look at the numbers: 12,345, 35,789, 24,608, 20,451, 48,093, 76,426, 100,000

Notice that 100,000 - 76,426 = 23,574

76,426 - 20,451 = 55,975

20,451 - ? = ?

48,093 - 24,608 = 23,485

35,789 - 12,345 = 23,444

23,485 and 23,444 are close; difference of 41.

23,574 and 23,485 differ by 89.

Not consistent.

Perhaps the gray bar is 23,485, and we box that as the answer, since it's the only logical choice from the right side.

And perhaps the 20,451 is a typo, and it should be 59,274, but since the problem asks for the gray bar, and from F = I + J, with F=48,093, J=24,608, so I=23,485.

So I'll go with that.

Final Answer: 23,485