Balance the Water Grid
平均分配：只移动一个杯子
Sixteen cups hold different amounts of water. Move exactly one cup, distribute all of its water among other cups, and make the four rows, four columns, and two long diagonals have equal totals.
Whole-unit version: pour all water from one donor in whole units and leave its original position empty. Assume other cups have enough capacity. Water heights use one scale within each board; the printed numbers give the exact amounts.
Read the rules and the grid correctly
The top labels identify columns, the labels at the left identify rows, row totals belong at the right, and column totals belong at the bottom. Try Stage 2 before consulting the live totals in Stage 5.
The challenge rules
- Choose exactly one donor cup.
- Move that cup and pour all of its water into one or more other cups.
- Do not add, lose, or create water.
- The donor cup’s original position must finish empty. Keep the empty donor cup outside the grid; all other cups stay at their original coordinates and may only receive water.
- Make all ten line totals equal: four rows, four columns, and two long diagonals.
Starting water grid
Row totals are shown at the right, column totals at the bottom, and diagonal totals below. The analysis board reveals its totals after you verify Stage 2; live totals are also available in Stage 5.
Check the rules
Calculate every starting line total
Do the arithmetic yourself. The page verifies your entries but does not print the totals in advance.
Use the coordinates
Main diagonal: R1C1, R2C2, R3C3, R4C4. Other diagonal: R1C4, R2C3, R3C2, R4C1.
Calculation reminders
- A row uses four cups from left to right.
- A column uses four cups from top to bottom.
- Each long diagonal uses four cups.
- The total water can be found by adding all four row totals.
calculate → record → verify
Enter your starting totals
Rows
Columns
Diagonals and total
Determine the common target without guessing
The four rows contain every cup exactly once. Use conservation and your own totals to determine the only possible common line value.
Reason from conservation
- Use your four row totals to find the amount of water in the whole grid.
- Imagine the four rows are balanced and equal.
- Divide the conserved whole-grid amount equally among those four rows.
- The columns and diagonals must match that same value.
Why this works
The rows partition all sixteen cups: no cup is omitted and no cup is counted twice. Moving water changes locations, but it does not change the whole-grid total.
conserved total ÷ number of equal rows = common line total
Enter your conclusion
Record a plan without being told the move
A plan is a prediction. Use the local strategy hints below if you need a starting point. The simulator will decide whether your chosen donor and pours actually work.
Choose one donor
Decide which cup you intend to move. You may revise your plan before solving.
Protect conservation
All water from the donor must be poured elsewhere. The original donor position must finish empty.
Verify globally
A move is successful only when all ten line totals—not just one row or one column—are equal.
Record your prediction
Strategy hints — open one at a time
1. Compare each line with your target
Which rows and columns have too much water? They must lose some.
2. Narrow the donor
Only the donor can lose water. Its position must allow every overfull row and column to lose water.
3. Plan recipients
After lifting the donor, list each line’s shortfall. A pour changes one row, one column, and sometimes a diagonal. Which shortfalls can one pour help together?
4. Check the whole grid
A row that is fixed can still cross an unfinished column. Check all four rows, all four columns, and both diagonals. Use Undo to revise a trial.
Solve the one-cup pouring challenge
Choose a donor, confirm it, and pour one unit at a time. The checker accepts any arrangement that satisfies every rule.
How the simulator works
- Click one cup and confirm it as the donor.
- The “water still in the moved cup” counter records water held in your donor cup, outside the grid. Its original position becomes empty. This counter is not an extra original of water.
- Click any other cup once for each unit you want to pour there.
- Watch the row totals on the right, column totals at the bottom, and diagonal totals below update immediately.
- Before a successful verification, Undo returns the most recent poured unit to the donor cup. Restart restores the original grid so you can choose a different donor.
- Press Verify. The page checks conservation and all ten line totals.
Verify the arrangement you created
This stage uses the arrangement you created in Stage 5. Record your own donor and pours; another valid arrangement may differ from the worked example.
Create a valid balanced arrangement in Stage 5 first.
Explain what makes a solution valid
The final check focuses on general reasoning and does not reveal any particular donor or pour pattern.
Validity conditions
- Exactly one donor cup was moved.
- The donor’s original position is empty.
- No water was added or lost.
- All four rows are equal.
- All four columns match the rows.
- Both long diagonals match them too.
Different methods may be valid
A static answer key can suggest that only one arrangement is possible. This page instead checks the mathematical conditions, so any valid arrangement is accepted.
rules satisfied → solution accepted
Strategy check
Water-Grid Balance Detective
You created and verified a valid one-cup balancing arrangement.
This optional badge does not change Chapter 8 mastery.
Teaching notes
The original includes a worked arrangement, but this student page intentionally omits it.
The digital checker accepts any integer-unit arrangement that preserves the total water, empties exactly one donor position, and makes the four row totals, four column totals, and two long-diagonal totals equal. During the interactive stage, all ten totals are displayed and update after every click.