Problem
Change the shape by moving coins
Six identical coins lie on the table in a cross. The vertical line holds 4 and the horizontal line 3. Only one coin may be moved. Make both lines hold 4 coins.
Reasoning · Grade 4-2 Games of Wit
Your answer
Watch out Sliding coins flat runs out of places, but nothing forbids stacking.
How to solve
Strategy Create a Physical Representation — This is a hands-on puzzle, so the honest way to attack it is to put six real coins on the table in the cross shape and start sliding them — guess a move, count both lines, and see what breaks. A few tries show that every slide that helps one line hurts the other, which is worth pausing over, because counting the places needed proves that no amount of sliding will ever work. That is the moment to change focus and ask what the puzzle actually demands: it says the coins lying in each line must number 4, not that all six coins must lie flat in six different places. Coins are solid objects that can be stacked, and that is the way out.
1STEP 1
Lay out the coins and count both lines
The two lines hold 4 and 3.
vertical = 4, horizontal = 3
Handling the actual objects makes the state of play obvious at a glance, and it is exactly the six coins in your hand that later suggest the trick.
4.G.A.1Create A Physical Representation2STEP 2
Try the obvious move and watch it fail
Moving one just swaps the 4 and the 3.
4 + 3 → 3 + 4 → 4 + 3
Guessing and checking a few moves is not wasted work — the fact that every guess trades one line against the other is itself the clue that something deeper is stopping you.
4.OA.A.3Guess And Check3STEP 3
Count the places and see that flat is impossible
A flat cross needs 7 places but there are only 6 coins.
4 + 4 - 1 = 7 ＞ 6
When guessing keeps failing, count what a winning position would demand — if the demand is bigger than what you have, you have proved the failures were not bad luck.
4.OA.A.3Change Focus Count The ComplementCounting the places shows a flat solution is impossible, because the two lines together would need more spots than exist.
Why?
Six coins can occupy at most six separate spots, so a demand for more spots than that cannot be met with one coin per place.
🧱More items than boxes forces a shared boxn boxes can hold at most n items one apiece, so item number n plus one has nowhere new to go and must share.
Why?
Showing the demand outruns the supply proves the failures were not bad luck, which is what forces the rules to be re-read.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
4STEP 4
Change focus: re-read what the puzzle asks
Nothing forbids stacking, so try going upwards.
3 + 3 - 1 = 5 places, 5 + 1 stacked = 6 coins
The 7-places count only applies while every coin must have its own spot; letting one spot hold two coins is not cheating, it is noticing a rule nobody actually made.
4.G.A.1Change Focus Count The Complement5STEP 5
Make the single move
Stack the bottom coin on the crossing coin.
1 + 2 + 1 = 4 each way, 1 + 2 + 1 + 1 + 1 = 6 coins
One stacked pair does the work of two separate places at once, which is precisely the single extra coin the horizontal line was short of.
4.OA.A.3Create A Physical Representation6STEP 6
Draw the arrow
Both lines then hold 4 coins.
bottom coin ⟶ crossing coin (stack on top)
An arrow from where a coin was to where it went is the clearest possible record of a move, and drawing the second circle is what tells the reader the coin is on top and not beside.
4.G.A.1Create A Physical RepresentationAnswer
stack the bottom coin onto the crossing coin
1 + 2 + 1 = 4
The count is right: 6 coins go in and 6 coins come out, none added and none removed, with the crossing place holding 2 and four other places holding 1 each — 2 + 1 + 1 + 1 + 1 = 6. Both lines read 4, as required. One move is also the fewest possible, since the starting picture does not already satisfy the condition and one move fixes it. The impossibility argument was checked exhaustively as well: listing every way to place 6 coins in the cells of a grid gives no arrangement at all that has both a row of 4 and a column of 4, which confirms that some kind of stacking is not just convenient but forced.
Takeaway
When no amount of sliding works, count the spaces you would need — and then check whether the puzzle ever really said the pieces had to stay flat.
- Lay out the coins and count both lines
- Try the obvious move and watch it fail
- Count the places and see that flat is impossible
- Change focus: re-read what the puzzle asks
- Make the single move
- Draw the arrow