## Tuesday, November 25, 2014

### The First Reconstruction Puzzle

A game of chess can be reconstructed from a record of the moves.

1. P-K4 P-K4
2. N-KB3 N-QB3
3. B-B4 B-B4
4. P-QN4 BxNP
5. P-B3 B-R4
6. P-Q4 PxP
7. O-O P-Q6
8. Q-N3 Q-B3
9. P-K5 Q-N3
10. R-K1 KN-K2

The above shows part of the record of a chess game. Provided you know the meaning you can follow the moves.

The same is true in the Game of Maths, by knowing what the recording means you can reconstruct the play. A form of puzzle in the game of maths is to leave the result from the recording and see if you can find it by reconstructing the game.

Approach the recording methodically and it is easier to obtain the result.

Examples

If a maths record looks like this 7 + 6 + (-5) + (-4) + 2 then it is easy to see that the + sign separates the positive and negative numbers and it is probably easy enough to see what cards were played. However mathematicians can be lazy and the record using addition and subtraction signs like this

7 + 6 – 5 – 4 + 2

Re-write using the list method, splitting onto a new row for each addition and subtraction. Do NOT write down the + signs but DO write down the – signs.

List
 7 where there is a 7 6 subtraction 6 -5 change to ⟶ (-5) -4 negative (-4) 2 2

Each row becomes a card that was part of the hand.

rearrange into black and white card groups.

 Record List 7 6 2 (-5) (-4) 15 (-9)   6

You can work out the result on the board with the black and white counters or perhaps it is possible to do it from the cards or the records.

 Record Maths 7 + 6 + 2 + (-5) + (-4) = 15 + (-9) = 6

Mixture of addition, subtraction and multiplication.

Remember a holder is shown using a x sign followed by a number.

5 + 3 x 4 – 6 x 2 – 3 x (-4) x (-2) – 5 x 7 may unfortunately also be recorded as
5 + 3 x 4 – 6 x 2 – 3 x –4 x –2 – 5 x 7

If this is so then whenever there is a – sign after a x sign put an opening bracket in front of the – sign and a closing bracket after the number that follows it

5 + 3 x 4 – 6 x 2 – 3 x (–4 )x (–2) – 5 x 7

Re-write using the list method, splitting onto a new row for each addition. Do the same for each subtraction provided the – sign is not in brackets.

Do NOT write down + signs but DO write down the – signs.

List
 5 where there is a 5 3 x 4 subtraction 3 x 4 -6 x 2 change lead ⟶ (-6) x 2 -3 x (-4) number to (-3) x (-4) -5 x 7 a negative (-5) x 7

Each row is a card, holder and card or chain of holders and card that was added to the hand.

Apply the flip rule for negative holders

You can work out the result on the board with the black and white counters or perhaps it is possible to do it from the cards or the records.

 Record List 5 3 x 4 6 x (-2) 3 x 4 x (-2) 5 x (-7) 5 12 (-12) (-24) (-35) 17 (-71) (-54)

 Record Maths 5 + 3 x 4 – 6 x 2 – 3 x (-4) x (-2) – 5 x 7 = 5 + 3 x 4 + (- 6) x 2 + (-3) x (-4) x (-2) + (-5) x 7 = 5 + 12 + (-12) + (-12) + (-35) = 17 + (-71) = (-54) = -54