5th Grade Math Puzzles - Free Printable

Let’s solve this step by step.

We start at 0 (as instructed: “Start the maze with zero”).

We must follow arrows from one hexagon to another, doing the math operation in each box as we go. We need to find two routes that end at 100, and then also find which route gives the highest total and which gives the lowest total — even if they don’t equal 100.

Wait — rereading the instructions:

> You have to finish the maze with a total of 100.
> There are two possible routes. Can you find them?
> Then: Which route has the highest total? The lowest total?

So actually, there may be more than two paths through the maze, but only two of them result in exactly 100 at the end. But then it asks for the highest and lowest totals among all possible routes? Or just comparing the two that reach 100? Let’s read again:

> There are two possible routes. Can you find them?
> Which route has the highest total? ...
> Which route has the lowest total? ...

It seems like after finding the two routes that give 100, we’re to compare those two? But that wouldn’t make sense because both would be 100. So probably, the question is: among all possible paths from START to END, some will give different totals. Two of them give 100. But we are also to find which path overall gives the highest number and which gives the lowest number — regardless of whether they hit 100.

But let’s look at the structure. It’s a grid of operations. From START, you can go right or down. Each choice leads to different operations. We need to trace all possible paths from START to END, calculate the final number for each, and see which ones equal 100, and also which is max/min.

First, let’s map out the grid positions.

Label rows and columns for clarity.

Think of the maze as a 4x4 grid of operation boxes, plus START and END.

Actually, looking at the diagram:

Row 1: START → +6 → x3 → 2
Row 2: +9 → x8 → ÷3 → +9
Row 3: x5 → -6 → +4 → x4
Row 4: ÷3 → x7 → -5 → END

And arrows connect:

From START:
- Right to +6
- Down to +9

From +6:
- Right to x3
- Down to x8

From x3:
- Right to ÷2
- Down to ÷3

From ÷2:
- Down to +9

From +9 (row2 col1):
- Right to x8
- Down to x5

From x8 (row2 col2):
- Right to ÷3
- Down to -6

From ÷3 (row2 col3):
- Right to +9
- Down to +4

From +9 (row2 col4):
- Down to x4

From x5 (row3 col1):
- Right to -6
- Down to ÷3

From -6 (row3 col2):
- Right to +4
- Down to x7

From +4 (row3 col3):
- Right to x4
- Down to -5

From x4 (row3 col4):
- Down to END

From 3 (row4 col1):
- Right to x7

From x7 (row4 col2):
- Right to -5

From -5 (row4 col3):
- Right to END

END is at bottom right.

So we need to find all paths from START to END, moving only right or down (since arrows only go right or down), and compute the result starting from 0.

Let’s list all possible paths. Since it’s a 4x4 grid of operations (but START is before row1, END after row4), we need to take exactly 3 rights and 3 downs? Wait, no.

Actually, from START to END, how many steps?

Looking visually: To get from top-left to bottom-right, in a 4-row by 4-column grid of operation nodes, you need to move right 3 times and down 3 times? But START is above row1, and END is below row4.

Actually, counting the moves:

From START, you enter the first row of operations. Then you traverse through 4 rows and 4 columns of operation boxes, ending at END.

Each path consists of choosing at each node whether to go right or down, until you reach END.

Since it's a grid, the number of paths is combinatorial, but let’s trace systematically.

I’ll denote each cell by (row, col), where row 1 to 4, col 1 to 4.

START is before (1,1). END is after (4,4).

Operations:

(1,1): +6
(1,2): x3
(1,3): ÷2
(1,4): nothing? Wait no — in row1, col4 is ÷2, and from there you can only go down to (2,4) which is +9.

Similarly, (2,1): +9
(2,2): x8
(2,3): ÷3
(2,4): +9

(3,1): x5
(3,2): -6
(3,3): +4
(3,4): x4

(4,1): 3
(4,2): x7
(4,3): -5
(4,4): END — but END is not an operation, it’s the finish. So when you reach (4,4), you stop? No, looking at the diagram, from (3,4) x4 you go down to END, and from (4,3) -5 you go right to END. Also from (4,2) x7 you go right to (4,3) -5, etc.

Actually, END is a separate node after (4,4)? In the diagram, the last operation before END is either from (3,4) x4 going down, or from (4,3) -5 going right. And (4,4) is labeled END, so perhaps (4,4) is the END node, and the operations are in the other cells.

Let me redefine:

The grid has 4 rows and 4 columns of operation hexagons, and START is to the left of row1 col1, END is to the right of row4 col4? No, in the diagram, START is top-left, then arrows go right and down, and END is bottom-right yellow hexagon.

Looking at connections:

- From START, you can go to (1,1) [+6] or to (2,1) [+9]? No, in the diagram, from START, arrow right to +6 (which is row1 col1), and arrow down to +9 (which is row2 col1). So actually, the first column has operations in row1 to row4, but START connects to row1 col1 and row2 col1? That doesn't make sense for a grid.

Perhaps it's better to think of the nodes as follows:

Let me sketch the connections explicitly.

Define nodes:

S = START

A = +6 (top row, second from left)

B = x3 (top row, third)

C = ÷2 (top row, fourth)

D = +9 (second row, first)

E = x8 (second row, second)

F = 3 (second row, third)

G = +9 (second row, fourth)

H = x5 (third row, first)

I = -6 (third row, second)

J = +4 (third row, third)

K = x4 (third row, fourth)

L = ÷3 (fourth row, first)

M = x7 (fourth row, second)

N = -5 (fourth row, third)

T = END (yellow)

Now connections:

From S:
- Right to A (+6)
- Down to D (+9)

From A (+6):
- Right to B (x3)
- Down to E (x8)

From B (x3):
- Right to C (÷2)
- Down to F (÷3)

From C (÷2):
- Down to G (+9) [only down, no right]

From D (+9):
- Right to E (x8)
- Down to H (x5)

From E (x8):
- Right to F (÷3)
- Down to I (-6)

From F (÷3):
- Right to G (+9)
- Down to J (+4)

From G (+9):
- Down to K (x4) [only down]

From H (x5):
- Right to I (-6)
- Down to L (÷3)

From I (-6):
- Right to J (+4)
- Down to M (x7)

From J (+4):
- Right to K (x4)
- Down to N (-5)

From K (x4):
- Down to T (END)

From L (÷3):
- Right to M (x7)

From M (x7):
- Right to N (-5)

From N (-5):
- Right to T (END)

Also, from C (÷2) down to G, and G down to K, K down to T.

From L down? No, L only goes right to M.

From H down to L, L right to M, M right to N, N right to T.

Now, we need all paths from S to T.

Let's list them.

Path 1: S -> A -> B -> C -> G -> K -> T

Operations: +6, x3, ÷2, +9, x4

Start with 0.

0 +6 = 6

6 x3 = 18

18 ÷2 = 9

9 +9 = 18

18 x4 = 72

Then to T. Total = 72

Not 100.

Path 2: S -> A -> B -> F -> G -> K -> T

S->A: +6 → 6

A->B: x3 → 18

B->F: 3 → 6

F->G: +9 → 15

G->K: x4 → 60

Total = 60

Path 3: S -> A -> B -> F -> J -> K -> T

S->A: +6 → 6

A->B: x3 → 18

B->F: ÷3 → 6

F->J: +4 → 10

J->K: x4 → 40

Total = 40

Path 4: S -> A -> B -> F -> J -> N -> T

S->A: +6 → 6

A->B: x3 → 18

B->F: ÷3 → 6

F->J: +4 → 10

J->N: -5 → 5

N->T: END, so total 5

Path 5: S -> A -> E -> F -> G -> K -> T

S->A: +6 → 6

A->E: x8 → 48

E->F: ÷3 → 16

F->G: +9 → 25

G->K: x4 → 100 ← This is 100!

Yes! One route to 100.

Path 6: S -> A -> E -> F -> J -> K -> T

S->A: +6 → 6

A->E: x8 → 48

E->F: ÷3 → 16

F->J: +4 → 20

J->K: x4 → 80

Total 80

Path 7: S -> A -> E -> F -> J -> N -> T

S->A: +6 → 6

A->E: x8 → 48

E->F: ÷3 → 16

F->J: +4 → 20

J->N: -5 → 15

Total 15

Path 8: S -> A -> E -> I -> J -> K -> T

S->A: +6 → 6

A->E: x8 → 48

E->I: -6 → 42

I->J: +4 → 46

J->K: x4 → 184

Total 184

Path 9: S -> A -> E -> I -> J -> N -> T

S->A: +6 → 6

A->E: x8 → 48

E->I: -6 → 42

I->J: +4 → 46

J->N: -5 → 41

Total 41

Path 10: S -> A -> E -> I -> M -> N -> T

S->A: +6 → 6

A->E: x8 → 48

E->I: -6 → 42

I->M: x7 → 294

M->N: -5 → 289

N->T: END, total 289

Path 11: S -> D -> E -> F -> G -> K -> T

S->D: +9 → 9

D->E: x8 → 72

E->F: ÷3 → 24

F->G: +9 → 33

G->K: x4 → 132

Total 132

Path 12: S -> D -> E -> F -> J -> K -> T

S->D: +9 → 9

D->E: x8 → 72

E->F: 3 → 24

F->J: +4 → 28

J->K: x4 → 112

Total 112

Path 13: S -> D -> E -> F -> J -> N -> T

S->D: +9 → 9

D->E: x8 → 72

E->F: ÷3 → 24

F->J: +4 → 28

J->N: -5 → 23

Total 23

Path 14: S -> D -> E -> I -> J -> K -> T

S->D: +9 → 9

D->E: x8 → 72

E->I: -6 → 66

I->J: +4 → 70

J->K: x4 → 280

Total 280

Path 15: S -> D -> E -> I -> J -> N -> T

S->D: +9 → 9

D->E: x8 → 72

E->I: -6 → 66

I->J: +4 → 70

J->N: -5 → 65

Total 65

Path 16: S -> D -> E -> I -> M -> N -> T

S->D: +9 → 9

D->E: x8 → 72

E->I: -6 → 66

I->M: x7 → 462

M->N: -5 → 457

Total 457

Path 17: S -> D -> H -> I -> J -> K -> T

S->D: +9 → 9

D->H: x5 → 45

H->I: -6 → 39

I->J: +4 → 43

J->K: x4 → 172

Total 172

Path 18: S -> D -> H -> I -> J -> N -> T

S->D: +9 → 9

D->H: x5 → 45

H->I: -6 → 39

I->J: +4 → 43

J->N: -5 → 38

Total 38

Path 19: S -> D -> H -> I -> M -> N -> T

S->D: +9 → 9

D->H: x5 → 45

H->I: -6 → 39

I->M: x7 → 273

M->N: -5 → 268

Total 268

Path 20: S -> D -> H -> L -> M -> N -> T

S->D: +9 → 9

D->H: x5 → 45

H->L: ÷3 → 15

L->M: x7 → 105

M->N: -5 → 100 ← Another 100!

Yes! Second route to 100.

Path 21: S -> D -> H -> L -> M -> N -> T — already did, 100.

Is there more?

What about S -> A -> E -> I -> M -> N -> T — did that, 289.

S -> D -> E -> I -> M -> N -> T — 457.

Now, also, from C, only down to G, then to K, to T — did that.

From G, only down to K.

From K, only down to T.

From L, only right to M.

From M, only right to N.

From N, only right to T.

Also, from H, can go right to I or down to L.

From I, right to J or down to M.

From J, right to K or down to N.

I think I have most paths.

But let me check if I missed any.

For example, S -> A -> B -> C -> G -> K -> T — did, 72.

S -> A -> B -> F -> G -> K -> T — 60.

S -> A -> B -> F -> J -> K -> T — 40.

S -> A -> B -> F -> J -> N -> T — 5.

S -> A -> E -> F -> G -> K -> T — 100 (path 5)

S -> A -> E -> F -> J -> K -> T — 80

S -> A -> E -> F -> J -> N -> T — 15

S -> A -> E -> I -> J -> K -> T — 184

S -> A -> E -> I -> J -> N -> T — 41

S -> A -> E -> I -> M -> N -> T — 289

S -> D -> E -> F -> G -> K -> T — 132

S -> D -> E -> F -> J -> K -> T — 112

S -> D -> E -> F -> J -> N -> T — 23

S -> D -> E -> I -> J -> K -> T — 280

S -> D -> E -> I -> J -> N -> T — 65

S -> D -> E -> I -> M -> N -> T — 457

S -> D -> H -> I -> J -> K -> T — 172

S -> D -> H -> I -> J -> N -> T — 38

S -> D -> H -> I -> M -> N -> T — 268

S -> D -> H -> L -> M -> N -> T — 100 (path 20)

Now, is there a path like S -> A -> E -> F -> G -> K -> T — already have.

What about S -> D -> H -> L -> M -> N -> T — have.

Also, from S -> A -> E -> I -> M -> N -> T — have.

I think that's all.

Now, the two routes that give 100 are:

1. Path 5: S -> A -> E -> F -> G -> K -> T : operations +6, x8, 3, +9, x4 → calculated as 6→48→16→25→100

2. Path 20: S -> D -> H -> L -> M -> N -> T : operations +9, x5, 3, x7, -5 → 9→45→15→105→100

Now, among all paths, what is the highest total? Looking at the calculations:

Path 16: S->D->E->I->M->N->T: 9→72→66→462→457 → 457

Path 10: S->A->E->I->M->N->T: 6→48→42→294→289 → 289

Path 14: S->D->E->I->J->K->T: 9→72→66→70→280 → 280

Path 19: S->D->H->I->M->N->T: 9→45→39→273→268 → 268

Path 8: S->A->E->I->J->K->T: 6→48→42→46→184 → 184

Path 17: S->D->H->I->J->K->T: 9→45→39→43→172 → 172

Path 11: S->D->E->F->G->K->T: 9→72→24→33→132 → 132

Path 12: S->D->E->F->J->K->T: 9→72→24→28→112 → 112

Path 1: S->A->B->C->G->K->T: 6→18→9→18→72 → 72

Etc.

The highest is Path 16: 457

Lowest? Let's see small numbers.

Path 4: S->A->B->F->J->N->T: 6→18→6→10→5 → 5

Path 7: S->A->E->F->J->N->T: 6→48→16→20→15 → 15

Path 13: S->D->E->F->J->N->T: 9→72→24→28→23 → 23

Path 18: S->D->H->I->J->N->T: 9→45→39→43→38 → 38

Path 9: S->A->E->I->J->N->T: 6→48→42→46→41 → 41

Path 6: S->A->E->F->J->K->T: 80

Path 3: S->A->B->F->J->K->T: 40

Path 2: S->A->B->F->G->K->T: 60

Path 1: 72

Path 4 is 5, which is low.

Is there lower?

Path 4: ends with -5, and before that was 10, so 5.

What about S->A->B->F->J->N->T: yes, 5.

Is there a path that goes to negative? Probably not, since we start at 0 and mostly add/multiply, but let's see.

For example, if we do division early, might get fractions, but in this case, all divisions seem to result in integers based on the paths.

In path 4: after F->J: +4 to 10, then J->N: -5 to 5.

Can we get lower than 5?

Look at S->A->B->F->J->N->T: 5

S->A->E->F->J->N->T: 15

S->D->E->F->J->N->T: 23

All higher.

What about S->A->B->C->G->K->T: 72

Or S->D->H->L->M->N->T: 100

Another path: S->A->E->I->J->N->T: 41

All seem above 5 except path 4.

But is there a path like S->A->B->F->J->N->T — that's 5.

What if we go S->D->H->L->M->N->T — 100.

Or S->A->E->F->G->K->T — 100.

Now, is there a path that ends with -5 and has smaller previous value?

For example, if we can get to J with less than 10, then -5 would be less than 5.

In path 4, we got to J with 10.

How did we get to J? From F->J: +4, and before that F was 6 (from B->F: ÷3 of 18).

If we come to F from elsewhere.

For example, S->D->E->F->J->N->T: F is 24 (from E->F: ÷3 of 72), then +4 to 28, then -5 to 23.

Higher.

S->A->E->F->J->N->T: F is 16 (from E->F: ÷3 of 48), +4 to 20, -5 to 15.

Still higher than 5.

Can we get to J with less than 10?

J is reached from F or from I.

From F: F->J is +4

From I: I->J is +4

So same operation.

To have low value at J, need low value before adding 4.

Minimum before adding 4: if we can get to F or I with 6, then +4=10, as in path 4.

Can we get to F or I with less than 6?

F is 3, so to get integer, input must be multiple of 3.

Similarly, I is -6, so depends.

For example, to get to F with value v, then v must be divisible by 3 for integer, but in the problem, probably assumes integer results, as per the paths.

In path 4, we had at B: 18, then ÷3 to 6 at F.

If we could get to B with less, but B is x3, so input to B must be such that after x3 is divisible by 3, which it always is, but to get low.

S->A->B: A is +6, so 0+6=6, then x3=18, minimum for B.

If we come to B from elsewhere? Only from A.

A is only from S or from? A is only entered from S right.

Similarly, D is from S down.

So minimum at B is 18.

Then F is 6.

Then J is 10.

Then N is 5.

If we come to I with low value.

I is -6, so if we have value w at I, then w = previous -6.

Previous could be from E or H.

E is x8, H is x5.

Minimum at E: if from A, 6*8=48, or from D, 9*8=72.

Minimum at H: from D, 9*5=45.

So at I: from E: 48-6=42, or 72-6=66; from H: 45-6=39. All high.

Then J: +4, so at least 43 or more.

Then N: -5, so at least 38.

Higher than 5.

So path 4 gives 5, which seems the lowest.

But is there a path that avoids large multiplications?

Another path: S->A->B->F->J->N->T: 5

What about S->A->B->F->J->K->T: 40, higher.

Or S->D->H->L->M->N->T: 100

Now, highest is path 16: 457

Confirm calculation:

S->D: +9 → 0+9=9

D->E: x8 → 9*8=72

E->I: -6 → 72-6=66

I->M: x7 → 66*7=462

M->N: -5 → 462-5=457

Yes.

Is there higher? Path 10: S->A->E->I->M->N->T: 6→48→42→294→289 < 457

Path 14: S->D->E->I->J->K->T: 9→72→66→70→280 < 457

Path 19: S->D->H->I->M->N->T: 9→45→39→273→268 < 457

So 457 is highest.

Lowest is 5 from path 4.

But the problem says "there are two possible routes" to 100, which we found: one is 100 via +6,x8,÷3,+9,x4 and the other via +9,x5,÷3,x7,-5.

Then it asks for highest and lowest total among all routes.

So highest is 457, lowest is 5.

But let me confirm if there is a path with lower than 5.

What about S->A->B->F->J->N->T: 5

Is there S->A->E->F->J->N->T: 15

Or perhaps S->D->E->F->J->N->T: 23

All higher.

What if we go S->A->B->C->G->K->T: 72

Or S->D->H->I->J->N->T: 38

No.

Another path: did I miss S->A->E->I->J->N->T: 41

All good.

So the two routes to 100 are:

Route 1: Start → +6 → x8 → ÷3 → +9 → x4 → End

Route 2: Start → +9 → x5 → 3 → x7 → -5 → End

Highest total: 457 (from Start → +9 → x8 → -6 → x7 → -5 → End)

Lowest total: 5 (from Start → +6 → x3 → ÷3 → +4 → -5 → End)

In the lowest, it's S->A->B->F->J->N->T: +6, x3, ÷3, +4, -5

Calculations: 0+6=6; 6*3=18; 18÷3=6; 6+4=10; 10-5=5. Yes.

Now, for the answer.

The questions are:

Which route has the highest total? The highest total is _____.

Which route has the lowest total? The lowest total is _____.

So we fill in the numbers.

Also, initially, it says "find the two routes" to 100, but the final answer part is for highest and lowest totals.

Looking back at the user's request, and the response format, since ExplainAnswer is false, TeachTopic false, GeneratePractice false, we only provide Final Answer.

But the Final Answer should be the answers to the blanks.

The worksheet has:

"Which route has the highest total? The highest total is _____."

"Which route has the lowest total? The lowest total is _____."

So we need to provide those numbers.

Additionally, it says "There are two possible routes. Can you find them?" but since ExplainAnswer is false, we don't explain, just give the final answer as per the blanks.

But the Final Answer section should contain the answers.

Perhaps we need to state the highest and lowest totals.

To be precise.

Since the instruction is to provide only the final correct answer after reasoning, and reasoning is done, now output.

But in the response, since no explanation sections are to be included, only Final Answer.

So:

Final Answer:
The highest total is 457.
The lowest total is 5.

But let me double-check if 5 is indeed achievable and correct.

Path: Start (0) → +6 → 6 → x3 → 18 → 3 → 6 → +4 → 10 → -5 → 5 → End. Yes.

And for highest: Start (0) → +9 → 9 → x8 → 72 → -6 → 66 → x7 → 462 → -5 → 457 → End. Yes.

And the two routes to 100 are confirmed.

So for the blanks:

Highest total is 457

Lowest total is 5

Now, in the worksheet, it might expect just the numbers.

But to match the format.

Perhaps write:

Final Answer:
457
5

But better to specify.

Since the question says "The highest total is _____." and "The lowest total is _____.", so in Final Answer, we can put:

The highest total is 457. The lowest total is 5.

But to keep it clean.

I think it's fine.

Note: In the path for lowest, we have ÷3 after x3, which is fine, 18÷3=6.

All operations result in integers, as expected.

So no issue.

Therefore, the answers are 457 and 5.