◆東京都 明 さんからの解答
【問題１】
○，○，×，×，×， ×，×，×，◎，◎， ×，×，×，×，◎， ○，×，×，×，×， ×，×，◎，◎，×【問題２】
基本的な配列は１通り。
回転（４種類）、裏返し（２種類）で８通りとなります。
【問題３】
まず豚２匹の基本配置を以下のように洗い出し、縦、横、斜めをつぶして、５箇所以上の空きができるものを拾い上げ、さらに一匹を追加して、最終的に５箇所の空きが残るものを拾いあげました。
結果、３つ（同型）を見つけました。
中央に１匹がいる場合（５通り）
中央になく、内周に少なくとも１匹の場合（２４通り）
外周のみに２匹の場合（２１通り）
論理的な決定法がわかりませんので洗い出しのプログラムを「（仮称）十進BASIC」で作ってみました。
これで計算すると、豚とライオンの関係は下記のようになります。
豚 ライオン 解の数 ０ ２５ １ １ １２ １６ ２ ７ ２４ ３ ５ ８ ４ ４ １８ ５ ３ ８ ６ ２ ２４６ ７ ２ ２４ ８ １（これ以後は時間がかかりすぎるので ・ ・ 実行していません。） １２ １ １６ １３ ０ ？ ・ ・ ２５ ０ １ライオン（または豚）の数が０となる場合 の解の数は、２５個から豚（またはライオン）の数を選ぶ 組み合わせの数となると思います。
【コメント】
広島県 清川 育男 さんからの情報によると、次のような結果になるそうです。
豚 ライオン 解の数
０ ２５ １
・ ・ ・
１ １３ ０
１ １２ １６
１ １１ １９２
１ １０ １０６４
１ ９ ３６００
１ ８ ８２８１
２ ７ ２４
３ ５ ８
４ ４ １８
５ ３ ８
６ ２ ２４６
７ ２ ２４
８ １ ８２８１
９ １ ３６００
１０ １ １０６４
１１ １ １９２
１２ １ １６
１３ １ ０
・ ・ ・
２５ ０ １
◆明さんのプログラム
DECLARE EXTERNAL SUB ZYUN
LET A=5 ! 配列数定義
LET W=3 ! 監視人数定義（豚の数）
DIM ARAY(A,A) ! 配置領域定義
DIM KANS(A,A) ! 監視人配置パターン領域定義
REM 基準配列の生成
LET HN=A*A ! 配置領域数
DIM N(HN)
MAT N=ZER
LET WW=W
FOR I=HN TO 1 STEP -1
IF WW>0 THEN
LET N(I)=1
LET WW=WW-1
END IF
NEXT I
REM 監視もれ数の最大値を求める
LET ZO=0 ! 監視もれ数
LET AC=0 ! 適合配列数
LET NO=0 ! 配置No
DO
MAT ARAY=ZER
LET NO=NO+1
FOR I=1 TO HN
IF N(I)=1 THEN CALL KANSI
NEXT I
CALL ZANSUU
IF ZN>ZO THEN
LET ZO=ZN
LET AC=1
ELSEIF ZN=ZO THEN
LET AC=AC+1
END IF
CALL ZYUN(HN,EX,N)
LOOP WHILE EX=1
PRINT "配列 : ";A;"×";A;" 監視人数 : ";W
IF ZO=0 THEN
PRINT "NOTHING"
STOP
END IF
PRINT "CHECK数 : ";NO;" 適合配列個数 : ";AC;" 監視もれ数 : ";ZO
REM 監視もれ数が最大の監視人配置を求める
MAT N=ZER
LET NO=0 ! 配置No
LET WW=W
FOR I=HN TO 1 STEP -1
IF WW>0 THEN
LET N(I)=1
LET WW=WW-1
END IF
NEXT I
DO
LET NO=NO+1
MAT ARAY=ZER
MAT KANS=ZER
FOR I=1 TO HN
IF N(I)=1 THEN
CALL KANSI
LET KANS(X,Y)=1
END IF
NEXT I
CALL ZANSUU
IF ZN=ZO THEN
PRINT
PRINT "CHECK No : ";NO
CALL OUTPUT
END IF
CALL ZYUN(HN,EX,N)
LOOP WHILE EX=1
PRINT
PRINT "END"
STOP
SUB KANSI ! 監視範囲の塗りつぶし
LET X=INT((I-1)/A)+1
LET Y=MOD((I-1),A)+1
FOR J=1 TO A
LET ARAY(X,J)=1
LET ARAY(J,Y)=1
NEXT J
LET DX=1
LET DY=1
FOR J=1 TO 4
CALL NANAME
LET DX=DX*DY
LET DY=-DY
NEXT J
END SUB
SUB NANAME ! 斜め方向の塗りつぶし
LET XX=X
LET YY=Y
DO UNTIL XX=0 OR XX=A+1 OR YY=0 OR YY=A+1
LET ARAY(XX,YY)=1
LET XX=XX+DX
LET YY=YY+DY
LOOP
END SUB
SUB ZANSUU !監視残数のCHECK
LET ZN=0
FOR I=1 TO A
FOR J=1 TO A
IF ARAY(I,J)=0 THEN LET ZN=ZN+1
NEXT J
NEXT I
END SUB
SUB OUTPUT ! 結果配列の出力
FOR I=1 TO A
FOR J=1 TO A
IF KANS(I,J)=1 THEN
PRINT "○ ";
ELSEIF ARAY(I,J)=0 THEN
PRINT "◎ ";
ELSE
PRINT "× ";
END IF
NEXT J
PRINT
NEXT I
END SUB
END
!順列の変更
EXTERNAL SUB ZYUN(K,EX,N())
DECLARE EXTERNAL SUB SEIRETU
LET EX=0
LET J=1
DO
IF J>=K THEN EXIT DO
IF N(K-J)<N(K-J+1) THEN
CALL SEIRETU(J,K,N)
LET L=1
DO
IF N(K-J)>=N(K-J+L) THEN
LET L=L+1
ELSE
LET B=N(K-J)
LET N(K-J)=N(K-J+L)
LET N(K-J+L)=B
LET EX=1
EXIT DO
END IF
LOOP
EXIT DO
ELSE
LET J=J+1
END IF
LOOP
END SUB
!配列の整列
EXTERNAL SUB SEIRETU(J,K,N())
LET D=K-J+1
LET E=K
DO WHILE D<E
LET B=N(D)
LET N(D)=N(E)
LET N(E)=B
LET D=D+1
LET E=E-1
LOOP
END SUB
◆千葉県 菜花子 さんからの解答
【問題１】
×，◎，◎，×，×， ×，◎，×，×，◎， ×，×，×，×，◎， ○，×，×，×，×， ○，×，×，○，× -------------------- ×，×，◎，◎，×， ◎，×，×，◎，×， ◎，×，×，×，×， ×，×，×，×，○， ×，○，×，×，○ -------------------- ×，×，◎，◎，×， ○，×，×，×，×， ×，×，×，×，◎， ×，×，×，◎，◎， ○，○，×，×，× -------------------- ×，◎，◎，×，×， ×，×，×，×，○， ◎，×，×，×，×， ◎，◎，×，×，×， ×，×，×，○，○ -------------------- ○，×，×，○，×， ○，×，×，×，×， ×，×，×，×，◎， ×，◎，×，×，◎， ×，◎，◎，×，× -------------------- ×，○，×，×，○， ×，×，×，×，○， ◎，×，×，×，×， ◎，×，×，◎，×， ×，×，◎，◎，× -------------------- ○，○，×，×，×， ×，×，×，◎，◎， ×，×，×，×，◎， ○，×，×，×，×， ×，×，◎，◎，× -------------------- ×，×，×，○，○， ◎，◎，×，×，×， ◎，×，×，×，×， ×，×，×，×，○， ×，◎，◎，×，×【問題２】
８通り
【問題３】
◆広島県 清川 育男 さんからの解答
●予想
(2n+1)*(2n+1)
| 豚（ライオン） | ライオン（豚） |
| 2n-1個 | (3n2-3n+4)/2個 |
5*5 1 ○,○,×,×,×, ×,×,×,△,△, ×,×,×,×,△, ○,×,×,×,×, ×,×,△,△,×, 2 ○,×,×,○,×, ○,×,×,×,×, ×,×,×,×,△, ×,△,×,×,△, ×,△,△,×,×, 3 ×,○,×,×,○, ×,×,×,×,○, △,×,×,×,×, △,×,×,△,×, ×,×,△,△,×, 4 ×,×,×,○,○, △,△,×,×,×, △,×,×,×,×, ×,×,×,×,○, ×,△,△,×,×, 5 ×,×,△,△,×, ○,×,×,×,×, ×,×,×,×,△, ×,×,×,△,△, ○,○,×,×,×, 6 ×,△,△,×,×, ×,×,×,×,○, △,×,×,×,×, △,△,×,×,×, ×,×,×,○,○, 7 ×,△,△,×,×, ×,△,×,×,△, ×,×,×,×,△, ○,×,×,×,×, ○,×,×,○,×, 8 ×,×,△,△,×, △,×,×,△,×, △,×,×,×,×, ×,×,×,×,○, ×,○,×,×,○, 7*7 1 ○,×,○,×,×,×,○, ×,×,×,×,△,×,×, ×,△,×,△,×,△,×, ×,△,×,×,△,×,×, ×,△,×,△,×,△,×, ×,×,×,△,△,×,×, ○,×,×,×,×,×,○, 2 ○,×,×,×,○,×,○, ×,×,△,×,×,×,×, ×,△,×,△,×,△,×, ×,×,△,×,×,△,×, ×,△,×,△,×,△,×, ×,×,△,△,×,×,×, ○,×,×,×,×,×,○, 3 ○,×,×,×,×,×,○, ×,×,△,△,△,×,×, ○,×,×,×,×,×,×, ×,×,△,×,△,△,×, ×,△,×,△,×,△,×, ×,×,△,×,△,×,×, ○,×,×,×,×,×,○, 4 ○,×,×,×,×,×,○, ×,×,△,△,△,×,×, ×,×,×,×,×,×,○, ×,△,△,×,△,×,×, ×,△,×,△,×,△,×, ×,×,△,×,△,×,×, ○,×,×,×,×,×,○, 5 ○,×,×,×,×,×,○, ×,×,△,×,△,×,×, ×,△,×,△,×,△,×, ×,×,△,×,△,△,×, ○,×,×,×,×,×,×, ×,×,△,△,△,×,×, ○,×,×,×,×,×,○, 6 ○,×,×,×,×,×,○, ×,×,△,×,△,×,×, ×,△,×,△,×,△,×, ×,△,△,×,△,×,×, ×,×,×,×,×,×,○, ×,×,△,△,△,×,×, ○,×,×,×,×,×,○, 7 ○,×,×,×,×,×,○, ×,×,×,△,△,×,×, ×,△,×,△,×,△,×, ×,△,×,×,△,×,×, ×,△,×,△,×,△,×, ×,×,×,×,△,×,×, ○,×,○,×,×,×,○, 8 ○,×,×,×,×,×,○, ×,×,△,△,×,×,×, ×,△,×,△,×,△,×, ×,×,△,×,×,△,×, ×,△,×,△,×,△,×, ×,×,△,×,×,×,×, ○,×,×,×,○,×,○, 9*9 1 ○,○,×,×,×,×,×,×,○, ×,×,×,△,△,△,△,×,×, ×,×,×,×,△,△,×,△,×, ×,×,△,×,×,×,△,△,×, ×,×,△,△,×,×,△,△,×, ×,×,△,×,△,×,×,△,×, ×,×,×,△,△,△,×,×,×, ×,○,×,×,×,×,×,×,○, ○,×,×,×,×,×,×,×,○, 2 ○,×,○,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ○,×,○,×,×,×,×,×,×, ×,×,×,×,△,×,△,△,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,×,×,△,△,×,△,×,×, ○,×,×,×,×,×,×,×,○, 3 ○,×,○,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ×,×,×,×,×,×,×,×,○, ×,△,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,△,△,×,×, ○,×,○,×,×,×,×,×,○, 4 ○,×,○,×,×,×,×,×,○, ×,×,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,△,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,×,△,×,×, ○,×,×,×,×,×,×,×,○, ×,×,×,△,△,△,△,×,×, ○,×,×,×,×,×,×,×,○, 5 ○,×,○,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,△,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,×,△,×,×, ×,×,○,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ○,×,×,×,×,×,×,×,○, 6 ○,×,○,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×, ×,×,×,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ○,×,○,×,×,×,×,×,○, 7 ○,×,×,×,×,×,○,×,○, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,×,×,×, ×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,○,×,○, 8 ○,×,×,×,×,×,○,×,○, ×,×,△,△,△,×,×,×,×, ×,×,×,×,×,×,○,×,○, ×,△,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,△,×,×,×, ○,×,×,×,×,×,×,×,○, 9 ○,×,×,×,×,×,○,×,○, ×,×,△,△,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,△,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,×,×, ○,×,×,×,×,×,○,×,×, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,×,×,○, 10 ○,×,×,×,×,×,○,×,○, ×,×,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,△,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,×,×, ○,×,×,×,×,×,×,×,○, ×,×,△,△,△,△,×,×,×, ○,×,×,×,×,×,×,×,○, 11 ○,×,×,×,×,×,○,×,○, ×,×,△,△,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,△,×, ○,×,×,×,×,×,×,×,×, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,○,×,○, 12 ○,×,×,×,×,×,×,○,○, ×,×,△,△,△,△,×,×,×, ×,△,×,△,△,×,×,×,×, ×,△,△,×,×,×,△,×,×, ×,△,△,×,×,△,△,×,×, ×,△,×,×,△,×,△,×,×, ×,×,×,△,△,△,×,×,×, ○,×,×,×,×,×,×,○,×, ○,×,×,×,×,×,×,×,○, 13 ○,×,×,×,×,×,×,×,○, ○,×,×,×,×,×,×,○,×, ×,×,×,△,△,△,×,×,×, ×,△,×,×,△,×,△,×,×, ×,△,△,×,×,△,△,×,×, ×,△,△,×,×,×,△,×,×, ×,△,×,△,△,×,×,×,×, ×,×,△,△,△,△,×,×,×, ○,×,×,×,×,×,×,○,○, 14 ○,×,×,×,×,×,×,×,○, ×,○,×,×,×,×,×,×,○, ×,×,×,△,△,△,×,×,×, ×,×,△,×,△,×,×,△,×, ×,×,△,△,×,×,△,△,×, ×,×,△,×,×,×,△,△,×, ×,×,×,×,△,△,×,△,×, ×,×,×,△,△,△,△,×,×, ○,○,×,×,×,×,×,×,○, 15 ○,×,×,×,×,×,×,×,○, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,○,×,×, ×,×,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,△,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,○,×,○, 16 ○,×,×,×,×,×,×,×,○, ×,×,×,△,△,△,△,×,×, ○,×,×,×,×,×,×,×,○, ×,×,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,△,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,×,△,×,×, ○,×,○,×,×,×,×,×,○, 17 ○,×,×,×,×,×,×,×,○, ×,×,△,△,△,△,×,×,×, ○,×,×,×,×,×,×,×,○, ×,×,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×, ×,△,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,×,×, ○,×,×,×,×,×,○,×,○, 18 ○,×,×,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×, ×,×,○,×,×,×,×,×,○, ×,×,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,△,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,△,△,×,×, ○,×,○,×,×,×,×,×,○, 19 ○,×,×,×,×,×,×,×,○, ×,×,×,△,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×, ×,△,×,△,×,△,×,△,×, ×,×,×,×,△,×,△,△,×, ○,×,○,×,×,×,×,×,×, ×,×,×,×,△,△,△,×,×, ○,×,○,×,×,×,×,×,○, 20 ○,×,×,×,×,×,×,×,○, ×,×,△,×,△,△,×,×,×, ×,△,×,△,×,△,×,△,×, ×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×, ×,△,△,×,△,×,×,×,×, ×,×,×,×,×,×,○,×,○, ×,×,△,△,△,×,×,×,×, ○,×,×,×,×,×,○,×,○, 予想の変更の可能性。 11*11 ○,×,○,×,×,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×,×,×, ○,×,○,×,×,×,×,×,○,×,○, ×,×,×,×,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×,×,×, ×,△,×,△,△,△,×,△,×,△,×, ×,×,×,△,△,△,△,×,×,×,×, ×,×,×,×,×,×,×,×,○,×,○, 9-32 ○,×,○,×,×,×,×,×,×,×,○, ×,×,×,×,△,△,△,×,×,×,×, ○,×,○,×,×,×,×,×,○,×,○, ×,×,×,×,△,×,△,×,×,×,×, ×,△,×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×,△,×, ×,△,×,△,×,△,×,△,×,△,×, ×,△,×,×,△,×,△,×,×,×,×, ×,△,×,△,△,△,×,△,×,△,×, ×,×,×,△,△,△,△,×,×,×,×, ○,×,×,×,×,×,×,×,○,×,○, 10-32
◆ 問題へもどる
◆ 今週の問題へ
数学の部屋へもどる